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\title%[Handedness and parity violation]
{Handedness, parity violation,\\ and the reality of space%
\thanks{To appear in Katherine Brading and Elena
Castellani (\emph{eds}), in preparation, \emph{Symmetries in
Physics: Philosophical Reflections} (Cambridge: Cambridge
University Press).}
}
\author%[O.\ Pooley]
{Oliver Pooley\\
  Exeter College, University of Oxford}
\date{19 December 2001; revised 31 July 2002}

\begin{document}

\maketitle

\section{Introduction}

This paper is about asymmetry rather than symmetry. More
specifically, its subject is the sort of spatial asymmetry
exemplified by human hands.  Hands lack any plane of mirror
symmetry. As a result they come in two varieties: left hands and
right hands. Similarly we talk of left-handed and right-handed
screws, left-handed and right-handed molecules, left-handed and
right-handed coordinate systems, or sets of axes, and so
on.\footnote{Note that it is not required that the objects lack
every sort of spatial symmetry.  Screws, for example, can have
(discrete) rotational symmetry if their threads are of the correct
pitch.}  Objects of opposite handedness that are otherwise
qualitatively identical are ``mirror images'' of each other. Kant
was the first philosopher to see something interesting in such
objects.  He called them \emph{incongruent counterparts}. They
clearly differ in some way.  For example, a glove which might be a
perfect fit for a right hand will not fit on its left-handed
incongruent counterpart.

A fundamental question with which I shall be concerned is:
\emph{in virtue of what does one such object differ from its
incongruent counterpart; what is the ground of the difference
between them}?  Kant first tackled this question in the context of
a philosophical dispute, still very much alive today, that goes by
the name of the substantivalist--relationalist debate
\citep{kant68}.\footnote{The debate concerns the ontological
status of space.  Its original protagonists were, on one side,
Newton and Samuel Clarke and, on the other side, Leibniz.
Substantivalists follow Newton in seeing space (or, in the context
of relativistic physics, spacetime) as some kind of substance.  It
is as real, and as fundamental, as the material objects and events
that exist in it. Relationalists follow Leibniz in denying that
space is a fundamental entity. They do not deny that material
objects are spatially extended. Nor do they deny that material
objects stand in determinate distance relations from one another.
But they hold that such facts are basic.  They deny that space
exists apart from such facts; that space has a reality of its own,
independent of material objects and their spatial properties.}

I will side with most---although admittedly not all---philosophers
in defending an account of incongruent counterparts according to
which they are \emph{intrinsically
identical}.\footnote{\citet{harper91} and \citet{walker78} are two
exceptions.} Moreover, I will defend a \emph{relational} account
of handedness according to which the difference between
incongruent counterparts is grounded in their relations to each
other and to other material objects. Kant thought that there were
reasons to reject such an account. Initially he concluded that the
difference between left and right hands did indeed come down to a
difference in their relational attributes, but that these involved
relations to ``universal space as a unity'' \citep[365]{kant68}.
Not long after reaching this conclusion, he also rejected this
substantivalist account of handedness. Instead he now believed
that the difference between incongruent counterparts was
fundamentally incomprehensible: that it could only be grasped in
perception, through a ``pure intuition,'' and not by any
``characteristic marks intelligible to the mind through speech''
\citep[396]{kant70}.

Accounts of handedness according to which incongruent counterparts
are intrinsically identical have recently faced criticism from a
new quarter.  In \emph{World Enough and Space-Time}, John Earman
argues that the fact that our world displays a \emph{lawlike}
left--right asymmetry poses a serious challenge to what I am
calling relational accounts of handedness.

This paper is primarily concerned with the implications of
Earman's argument.  However, it is useful to consider issues
raised by Kant's argument first, for these help isolate the real
interpretative difficulties posed by parity violation.

\section{Incongruent Counterparts}

Imagine that you are given a model of a left hand and a perfect
mirror-image (i.e.\ right-handed) duplicate of it. The distance
between the tip of the thumb and the index finger will be the same
for both hands. Similarly, the angles that the thumbs make to the
planes of the palms will be identical in both cases. The two hands
are perfectly identical in terms of the distances between their
corresponding parts.  Kant's way of making this point was to note
that a complete description of one hand in terms of the positions
of the parts relatively to one another will also be true of its
mirror-image \citeyearpar[370]{kant68}.  For this reason he called
them \emph{counterparts}.

Yet despite this similarity, the two hands are nevertheless
\emph{incongruent}: they cannot be made to coincide---they cannot
be superposed---by any rigid motion.  Kant's own description of
the incongruence runs: ``the limits of the one cannot also be the
limits of the other'' \citeyearpar[369]{kant68}. A little later he
makes the same point by noting that the surface which encloses the
physical space of one hand ``cannot serve as a boundary to limit
the other, no matter how that surface be twisted and turned''
\citeyearpar[371]{kant68}. It is worth stressing that the relevant
notion of possibility here is not that of physical possibility. It
is physically impossible to superpose a left hand and its perfect
\emph{left-handed} duplicate if they are both solid material
objects. Rather we must abstract from such physical limitations
and consider whether it is \emph{mathematically} possible for the
distances between the two objects to be changed continuously in
such a way that the two objects eventually coincide.  By
restricting ourselves to \emph{rigid} motion, we are only
considering changes of the total set of distances that preserve
the internal distances between the parts of the two objects.

It is time to state a question that will concern us for much of
this section: \emph{how is it possible that two objects which are
counterparts can nevertheless be incongruent}? Answers to this
question fall into two categories, within which there are further
divisions.  One's answer might involve the claim that left- and
right-handed varieties of an object differ intrinsically, and then
go on to exploit these differences in explaining their
incongruence.  Within this category one might view these intrinsic
differences as primitive and unanalyzable (Kant's later
transcendental idealist position is perhaps of this type), or one
might view the intrinsic differences as resulting, for example,
from a difference in the way the parts of the objects are related
to each other.\footnote{Van~Cleve calls this latter position
``internalism'' \citep[22]{vancleve91}.  Clearly such relations
must include more than just the relative distances and relations
that are reducible to them.}  Alternatively one's answer might
involve the claim that incongruent counterparts do not differ
intrinsically in any way, and then seek to explain their
incongruence in terms of something that is external to each object
taken by itself.  Kant's earlier substantivalist position is of
this type: the difference is to be explained in terms of the
objects' different relations to substantival space. But so too is
the most economical, purely relational account according to which
incongruence is explicable using only the resources of relative
distances.

Kant's transcendental idealism aside, the view that left-handed
and right-handed objects differ intrinsically has not been
popular.  There are at least three strong objections to such a
view, one of a general nature and two that are specific. The
general objection is that it is entirely unclear that we have any
conceptual grasp of what such intrinsic differences
involve.\footnote{This objection does not tell against Kant's
later position, for he was at pains to deny that we have a
conceptual understanding of the left--right distinction; it is
supposedly only grasped in experience.} In particular, no one has
provided an illuminating account of how such intrinsic differences
connect with incongruence, or with relevant practical abilities;
for example, that I can tell without difficulty of a left hand
that it is left-handed and distinguish it from a right hand.

The two specific objections are variations on a
theme.\footnote{The following type of considerations are
emphasized by \citet[51--3]{nerlich94}, but see also
\citet[8]{frederick91} and \citet[22--3]{vancleve91}.} The first
exploits the fact that two hands will never be incongruent if
embedded in a non-orientable space.\footnote{Or, to adopt
relationalist language, if the spatial relations between them are
such that the lowest-dimensional spaces in which they are
embeddable are all non-orientable.  Some believe that failure to
provide an account of the orientability of space might show that
relationalism is ultimately untenable.  I disagree, for reasons
elaborated on below, pp.~\pageref{enantio}ff.} So can hands in
such spaces instantiate different intrinsic properties (primitive
or otherwise), or not? If they can, one will be able to move a
`left-handed' object into the space occupied by a `right-handed'
one. Does a hand originally exemplifying the property of being
`left-handed' come to lose this basic property and acquire that of
being `right-handed' merely as a result of such a motion? Although
they are said to differ in some primitive property, this property
appears redundant, making no contact with any other spatial facts
about the hands. But if hands in such spaces cannot instantiate
the primitive properties of being left- or right-handed, then how
is one meant to understand the dependence of whether or not an
object can instantiate such a property on the type of space in
which it is embedded?

The other objection runs along the same lines, but this time
exploits the fact that the incongruence of two hands in part
depends on the dimensionality of the space in which they are
embedded. One could move a left hand into the space occupied by a
right hand if there was an extra spatial dimension through which
it could be moved,\footnote{Although string theorists would have
us believe there are in fact six or more such dimensions, they are
`too small' to permit the required motion.} just as the letter
``F'' can be brought into coincidence with its mirror image on the
page, \reflectbox{F}, if one is allowed to lift it off the page.
Does whether or not hands have primitive left-handed or
right-handed properties depend on the dimensionality of space?

These considerations give us more that enough reason to see
whether incongruence can be explained without recourse to the
postulation of intrinsic differences.  I start by considering the
purely relational account. Can we get by merely with relative
distances?

The incongruence of left and right hands shows that they differ in
some respect.  It is surely reasonable to call this difference a
purely \emph{spatial} difference.  That left and right hands match
in terms of the distances between their parts shows that this
difference not grounded in \emph{these} distances.  However, the
relationalist is not committed to view that every spatial
difference between two objects supervenes on a difference in the
spatial arrangement of their parts.  He can also appeal to the
distance relations that hold between the two objects, and between
them and other objects.  Once this is acknowledged, there would
seem to be no reason why the relationalist cannot view the
incongruence of two counterparts as grounded in such external
relations.

Now nothing in my original definition of incongruence precluded
relationalism.\footnote{The definition deliberately does not
follow others that can be found in the literature; see e.g.
\citet[Chap.~7]{earman89}, \citet{brighouse99} and
\citet{huggett00}.} In fact, one might even think that
incongruence has been defined in purely relational terms: rigid
motion is defined in terms of the constancy of the distances
between the parts of an object and coincidence (the occupancy of
the same boundaries) is defined in terms of the distances between
(the corresponding parts of) the two objects. Even when we seek
the \emph{ground} of such incongruence, it seems that the
relationalist has no reason to be embarrassed.  It is simply a
mathematical fact, \emph{and a comprehensible one}, that, when
constrained to obey the algebraic relationships of Euclidean
geometry, some numbers (the possible distances between two
congruent counterparts) can be continuously altered so as to
vanish while others (the distances between two incongruent
counterparts) cannot.

The claim that the relationalist can not only accommodate but can
\emph{explain} incongruence is significant.  As already noted,
Kant came to the view that the incongruence of counterparts was,
in a certain sense, fundamentally incomprehensible. He thought
that it could be grasped only in experience. In his Inaugural
Dissertation he writes:
\begin{quote}
%Which things in a given space lie in one direction and which
%things incline in the opposite direction cannot be described
%discursively nor reduced to characteristic marks of the
%understanding by any astuteness of the mind.  Thus,
between solid bodies which are perfectly similar and equal but
incongruent, such as left and right hands (in so far as they are
conceived only according to their extension), or spherical
triangles from two opposite hemispheres, there is a difference, in
virtue of which it is impossible that the limits of their
extension should coincide -- and that, in spite of the fact that,
in respect of everything which may be expressed by means of
characteristic marks intelligible to the mind through speech, they
could be substituted for one another.  It is, therefore, clear
that in these cases the difference, namely the incongruity, can
only be apprehended by a certain pure intuition.
\citep[396]{kant70}
\end{quote}
But this is simply a \emph{non sequitur}.  Suppose that
incongruent counterparts \emph{are} intrinsically identical; that
they do not differ \emph{in themselves} in any way.  So \emph{a
fortiori} we cannot, restricting ourselves to just the internal
distances between the parts of two hands, understand or explain
the hands' incongruence.  Nevertheless we can both understand and
explain their incongruence in terms of the different ways any two
hands can be related to each other. It just does not follow from
the fact that we cannot intellectually grasp an \emph{intrinsic}
difference between left and right hands that we can have no
intellectual grasp of the basis of their incongruence, or that
this incongruence is manifest only in experience.

Those whose intuitions lead them still to side with Kant at this
point need to respond to the following challenge: Suppose, for the
sake of argument, that the relationalist is correct in asserting
that all spatial facts are reducible to facts about relative
distances between material objects. The relationalist will insist
that any two coexisting hands stand in some quite determinate
distance relations from one another. How can it be denied that the
possibility of incongruent counterparts is already
secured?\footnote{I have been tacitly assuming that the relative
distances involved are those of an infinite $N$-dimensional
Euclidean space.  Things are obviously more complicated when one
considers more general sets of relative distances.  For some
discussion, see \citet[Chap.~6]{pooley02}.} When things are put
this way, surely the burden of proof is now on someone who wishes
to assert that the incongruence or otherwise of the two hands is
not determined \emph{despite} the various facts about the
distances between the them.

So far we have seen that relative distances alone are sufficient
to ground the incongruence of two handed objects that are
otherwise identical.  However, there are two other questions about
handedness that have exercised philosophers. The first is: in
virtue of what is an object handed? \label{left?}The second is:
what accounts for the particular handedness of a handed
object---what makes it, say, a \emph{left} hand?

My initial characterization of a handed object suggested that its
key feature was that it lacked any plane of mirror symmetry. This
is, again, a feature highlighted by Kant, who noted that a handed
object cannot consist of ``two halves which are symmetrically
arranged relatively to a single intersecting plane''
\citep[370]{kant68}. One might think that this is a characteristic
that is reducible to facts concerning the relative distances
between the parts of the object.\footnote{Carol Brighouse,
however, worries that talk of a plane of symmetry and of lines
intersecting it at right-angles is not obviously relationally
acceptable \citep[56--8]{brighouse99}. The relationalist strategy
that I outline below sidesteps Brighouse's worries.} However, as
Nerlich notes, more is needed.

He defines an \label{enantio}\emph{enantiomorph} as follows:
\begin{quote}
each reflective mapping of [an enantiomorph] differs in its
outcome from every rigid motion of it. \citep[51]{nerlich94}
\end{quote}
Otherwise the object is a \emph{homomorph}. Nerlich's principal
contention is that ``whether a hand\ldots is enantiomorphic or
homomorphic depends on the nature of the space it is in.  In
particular it depends on the dimensionality or the orientability,
but in any case on some aspect of the overall connectedness or
topology of the space'' \citeyearpar[53]{nerlich94}. Nerlich's
claim, then, is that an object's being handed is not reducible to
facts about the relative distances between material objects.  It
also depends on the dimensionality and orientability of the space
in which the object is embedded.

Even if one were to agree with Nerlich about this, it is not clear
that this observation can be used as an argument against a
relational account of space, and for two reasons.  The first is
that we should ask why the relationalist about space is under any
obligation to offer an equivalent, relationally pure definition of
enantiomorphy.  He believes that spatial facts are exhausted by
the catalogue of relative distances between material points (and
the fact that these must obey certain constraints).  We have seen
that this is enough to allow for the possibility of incongruent
counterparts. If it turns out that substantivalism underwrites
properties, such as enantiomorphy, which are not well-defined by
the lights of the relationalist's ontology, then so much the worse
for enantiomorphy.  Nothing in our experience of objects such as
hands forces us to admit the existence of such additional
properties, just as (so the relationalist would like to maintain)
nothing in our theorizing about motion forces us to admit the
reality of space.

Secondly, it has yet to be shown that the relationalist cannot
provide a definition of enantiomorphy.  Nerlich's observations
might suggest that the relationalist needs to come up with a
surrogate definition of orientability, and this is indeed the
strategy that most have pursued \citep{brighouse99,huggett00}.
Unfortunately for the relationalist, it has not been entirely
successful.  For example, although an object's being multiply
related is a necessary condition of its being embeddable in a
non-orientable space (for all non-orientable spaces are multiply
connected), it is not a sufficient condition.  Kant's own example
of triangles on a sphere is precisely an example of multiply
related yet enantiomorphic figures.  Huggett suggests that what is
needed is a ``general representation theorem of the form `space
$S$ is \emph{orientable} iff relations of type $\underline{\ \ \ \
\ \ }$ are instantiated'~'' \citep[225]{huggett00}. No such
theorem has been forthcoming. The relationalist also needs an
account of the dimensionality of space.

I wish to propose that the relationalist has a way of sidestepping
some of these difficulties.  First, in order to be able to
\emph{exploit} (rather than explain) the fact\footnote{If it is a
fact.  This is something that the relationalist will want to
prove.  However, the mere possibility that it is a fact is enough,
at this stage of the dialectic, to save the relationalist. The
onus is now on the substantivalist to prove that the exact nature
of an object's multiple relatedness does \emph{not} fix the
orientability of the lowest-dimensional embedding spaces.  Thanks
to Jeremy Butterfield and Carl Hoefer for saving me from
overstating the relationalist's case.} that the exact nature of an
object's multiple relatedness (if it is multiply related) can
determine whether or not the spaces in which it is embeddable are
orientable or not, the relationalist does not need to have the
type of representation theorem to which Huggett alludes.  Second,
as Huggett notes, the relationalist can talk freely of embedding
the particular relative distances between the parts of some
material object in a space, so long as the operation is understood
to be a purely mathematical exercise \citep[224]{huggett00}.

So now suppose, additionally, that the relationalist has an
account of the dimensionality of space.\footnote{This might be
fixed by the adoption of some specific relational dynamical
theory. Relational theories typically simply assert, via the
choice of some relative configuration space for example, that the
relative distances between material objects are constrained to be
embeddable in, say, a Euclidean space of no more than three
dimensions; see, for example, \citet{barbourbertotti82}.  Note
that this is also likely to fix the orientability of space
directly.} He can then define the enantiomorphy of a material
object by adopting Nerlich's definition, but now with respect to
all abstract embedding spaces of the specified dimension. For
example, if it is the case that according to an empirically
adequate relational theory space has three dimensions, the
relationalist can claim that an object is an enantiomorph iff,
with respect to every possible abstract 3-dimensional embedding
space, each reflective mapping of the object differs in its
outcome from every rigid motion of it.  On this definition, planar
objects count as homomorphs as do 3-dimensional hands that are
multiply related so as to be embeddable only within non-orientable
3-dimensional spaces.  Hands that are embeddable only within
orientable 3-dimensional spaces count as enantiomorphs, even
though they are, of course, embeddable in spaces of higher
dimensions.

One suspects that Nerlich himself would be prepared to grant much,
if not all of this \citep[see, e.g.,][61--2]{nerlich94}. He would
still see the phenomena of handedness and enantiomorphy as
supporting substantivalism for two reasons.  One, which takes us
beyond our topic, is that he already sees the relationalist's
employment of brute, unmediated, spatial relations as suspect
\citep[see, especially,][23--33]{nerlich94}.  The second is that
he believes that the substantivalist account of enantiomorphy is
simply more illuminating.  The relationalist account, to the
extent that it explains anything at all, does so on the back of
the substantivalist one:
\begin{quote}
\ldots the orientability of space does determine the handedness of
hands, for it determines which paths there are in a space which a
hand might take.  It is a genuinely explanatory idea.  Spatial
relations, I suggest, explain enantiomorphy only by way of
entailing the orientability of the containing space, and it is
through that understanding that we come to grasp handedness.
\citep[67]{nerlich94}
\end{quote}

This, however, is disputable on two grounds.  First the
relationalist account of incongruence (if not of Nerlich's
definition of enantiomorphy) does \emph{not} appeal to facts about
embedding spaces. All it deals in are facts about relative
distances and about what changes of relative distances are
possible, and ``possible'' here need not be thought of as
constrained by an embedding space.

Second, the facts that the substantivalist appeals to are really
of exactly the same type. Ultimately the spatial relations between
the parts of space are either simply assumed to be, or are
stipulated to be, constrained in certain ways.  That the
substantivalist believes that relations between material points,
and the possible motions of those points, are mediated and
constrained in virtue of those points being located at various
spatial points is, quite simply, a distraction. How is Nerlich to
explain the incongruence of two hand-shaped regions of
\emph{space}? Space and its shape might be easier to
\emph{picture} than algebraic facts about distances, but the idea
that it is more explanatory is illusory.  For even in this case,
the facts that ultimately explain are precisely algebraic facts
about distances.

I now return to the second question raised on
page~\pageref{left?}: what accounts for the particular handedness
of a handed object? To see that, once again, nothing more than
relative distances is required, it will prove useful to consider
the anti-relationalist argument of Kant's 1768 paper.

In this paper, Kant explicitly characterizes his aim as that of
providing a ``clear proof that: \emph{Absolute space,
independently of the existence of all matter\ldots has a reality
of its own}'' \citeyearpar[366]{kant68}. In other words he sets
out to vindicate Newton's substantivalist conception of space over
Leibniz's relationalist conception.\footnote{Sometime prior to
1768, Kant is generally acknowledged to have held a Leibnizian,
relational view of space. This, however, is a matter of some
controversy amongst Kant scholars.  Some argue that he is better
seen as advocating some kind of compatibilism. Things are further
complicated by the fact that, as already discussed, just two years
after apparently arguing for a Newtonian view of space, Kant
published his first ``critical'' work, the Inaugural Dissertation
of 1770, is which he rejects \emph{both} substantivalism and
relationalism, arguing instead that space is in some sense `in
us', a form of our intuition.  The seeds of Kant's transcendental
idealism about space are already discernible in the 1768 essay.
However, the extent to which incongruent counterparts by
themselves led Kant to transcendental idealism is again a matter
of some controversy.

What should be stressed for the purposes of the present discussion
is that it is evident that by 1768 Kant believed that his argument
from incongruent counterparts provided a decisive reason to reject
a purely relational account of handedness. Nowhere in his
subsequent writings does Kant retreat from this claim.}  His
argument does not challenge the claim that the relationalist can
account for the \emph{incongruence} of left and right hands.
Rather it suggests that the difference between left and right goes
beyond the relational facts so far cited.

After rehearsing the various definitional facts about incongruent
counterparts and after noting that their incongruence cannot be
grounded in a difference in how their parts are related, Kant
makes the following claim:
\begin{quote}
\ldots imagine that the first created thing was a human hand. That
human hand would have to be either a right hand or a left hand.
The action of the creative cause in producing the one would have
of necessity to be different from the action of the creative cause
producing the counterpart.
\end{quote}
Kant rightly notes that this is incompatible with relationalism:
\begin{quote}
\ldots there is no difference in the relation of the parts of the
hand to each other, and that is so whether it be a right hand or a
left hand; it would therefore follow that the hand would be
completely indeterminate in respect of such a property.  In other
words, the hand would fit equally well on either side of the human
body; but that is impossible.
\citeyearpar[371]{kant68}.
\end{quote}

How should the relationalist respond to this particular challenge?
He can simply deny Kant's initial premise, that every hand in an
otherwise empty universe is necessarily either a right or a left
hand.  Certainly Kant is wrong to suppose that the lone hand's
being of indeterminate handedness entails the absurdity that it
can fit on \emph{both} sides of a human body.  For suppose that
one is given a relational description of a hand and also a
relational description of a handless human body that has various
internal asymmetries involving the heart and other organs.  One
might then ask on which side of this body does the hand (properly)
fit: the side on which the heart is, or on the other side?

The relationalist certainly should not answer ``both''. Rather he
will deny that the question makes sense independently of a
specification of the relative distances between the body and the
hand. There are two incompatible ways in which a body satisfying
the relational description and the lone hand could coexist in a
single universe. According to one such way, the hand will fit on
the side of the body that the heart is on. According to the other
way, the situation is reversed; the hand fits on the other side of
the body to that on which the heart is. But either way, the hand
will fit determinately on one, and only one, side of the body. And
which side it fits is determined by the distances between the hand
and the various parts of the body, i.e.\ by purely relational
facts.

Although the relationalist's contention that the difference
between left and right hand supervenes on the distances between
them and between other material objects does not entail a patent
absurdity, one might still wonder whether it is not in tension
with our evident ability to recognize, for example, left hands as
\emph{left}. However, a little reflection suggests that the
account the relationalist must give of our practical abilities and
linguistic practices---of how we teach the meanings of ``left''
and ``right'' and of the fact that we are often prone to confuse
left and right---is far more plausible than any account which
postulates our recognizing an intrinsic difference between
incongruent counterparts or recognizing that the hand bears some
relation to (invisible) space. In fact, so far as I know, no one
has attempted to give a genuine account of our abilities that
postulates our recognizing either of these things.

The basic elements of a relational account have been outlined many
times.\footnote{See, for example, \citet[Ch.~7, \S2]{earman89},
\citet[Ch.~17]{gardner90}, \citet[209--12]{huggett99},
\citet[\S3]{hoefer00} and \citet[\S3.3]{saunders00a}. These
accounts, of course, differ from each other, and from my own, in
minor ways.} We have seen that, despite holding that left and
right hands are intrinsically identical, the relationalist will
also acknowledge that they fall into two equivalence classes
defined, roughly speaking, by the relation of congruence. But it
is then straightforward to understand how a practice of
distinguishing members of these classes might involve all the
hands of one class being given one `name' (``left''), and all
hands of the other being given another (``right''). Causal links
between the speakers who are party to this practice, and between
the speakers and actual hands, will ensure that the practice
remains consistent. Together with an ability to recognize a hand
as congruent or incongruent to hands with which one has previously
been presented and has been told are left or right, these causal
links are all that are required.

According to the relationalist account, therefore, the \emph{only}
facts about a left hand that make it left, is the fact that
\emph{we call it ``left''}, that it is congruent to every other
hand that we in fact call ``left'' and incongruent to every hand
that we call ``right''.\footnote{The question ``in virtue of what
is a left hand left'' is thus rather misleading.  I should perhaps
stress that my favoured relational account of handedness is not
part of a general nominalism according to which the instances of
any general term ``X'' have in common only the fact that we call
them all ``X''.  It is only the left--right contrast, not
handedness \emph{per se}, that is purely nominal.} Such an account
of the meanings of ``left'' and ``right'' is, of course, very
close to a causal theory of reference for proper names. And in
certain respects the terms ``left'' and ``right'' are very much
like names. What was it \emph{about} Immanuel Kant, for example,
that made it correct for his contemporaries to call him ``Immanuel
Kant''? \emph{Nothing}, other than the fact that he was actually
known as ``Immanuel Kant'', that there was a practice of calling
him ``Immanuel Kant'' and so on.

Can this really be all there is to the left--right distinction? I
believe that it is. Such a point of view receives indirect support
from what Jonathan Bennett calls the \emph{Kantian Hypothesis}
\citep{bennett70}.\footnote{The reason for attributing this
hypothesis to Kant is Kant's insistence, noted earlier, that the
difference between left and right cannot be made intelligible
through concepts.} This is the claim that chiral terms such as
``left'' and ``right'' cannot ultimately be explained without
ostensively demonstrating, for example, a left hand. Various
chiral terms can be explained in terms of each other.  For
example, one can define ``left'' in terms of ``clockwise'' and
other, related, notions. But to break out of a rather \emph{tight}
circle, one must ultimately \emph{show} what one means by
``clockwise'' or by ``left.'' Non-chiral words will never be
enough.

This thesis can be put in the form of a predicament that Martin
Gardner calls the \emph{Ozma problem} \citep[Chap.~18]{gardner90}.
Suppose that we are in radio contact with some extra-galactic
civilization. Gardner's Ozma problem is: ``Is there any way to
communicate the meaning of ``left'' by a language transmitted in
the form of pulsating signals?  By the terms of the problem we may
say anything we please to our listeners, ask them to perform any
experiment whatever, with one proviso: \emph{There is to be no
asymmetric object or structure that we and they can observe in
common}'' \citeyearpar[167]{gardner90}. If Bennett's `Kantian
hypothesis' is correct, we cannot manage the task without some
asymmetric observable object in common with our alien friends.
Appealing to the side of the body on which the heart is won't
help, for example, because the alien hearts, if they have hearts,
might be on the right.\footnote{Actually, as will become apparent,
we have known since the 1950s that we could exploit the fact that
the laws of nature violate parity. One might worry that, since we
are communicating through photons, we cannot rule out the
possibility that our alien correspondents live in an
\emph{anti-matter} galaxy.  Were the laws CP invariant, their
following our instructions and carrying out an experiment
illustrating parity violation would lead them to conclude that
``left'' meant right. Fortunately we can appeal to CP violating
experiments to overcome this potential problem.  Moreover TCP
invariance is not a problem because our communicating at all
presupposes that we agree about ``before'' and ``after''!} If the
mechanism in virtue of which the terms ``left'' and ``right''
refer is indeed what I have suggested it is, the difficulty of
explaining their meanings within the constraints of the Ozma
problem are readily understandable.

Although Kant concludes in favour of substantivalism in his 1768
essay, he appears to do so very much by default. There is no
explanation in his essay of \emph{how} substantival space is able
to ground that which relationalism supposedly cannot: the
incongruence of counterparts.  \emph{If} substantivalism and
relationalism represent two genuinely exhaustive alternatives,
then an argument against one would be an argument for the other.
But as I noted earlier, Kant quickly came to the view that they
are not jointly exhaustive, and instead opted for the
\emph{tertium quid} of transcendental idealism.

However, \citet{hoefer00} has recently pointed out that there is
one way in which the postulation of substantival space \emph{can}
be used to secure Kant's intuition that a hand in an otherwise
empty universe is necessarily either a left or a right hand. One
is to imagine that the universe contains a single hand and that
the space in which the hand exists is the substantival space of
our \emph{actual} world. One would then appear to be able to
appeal to facts of the following sort: in the imagined possible
world, the lone hand is either determinately congruent to the
hand-shaped region of space that is actually and currently
occupied by my left hand, or it is determinately congruent to the
hand-shaped region of space that is actually and currently
occupied by my right hand. In the first case the lone hand is
left-handed, in the second it is right-handed
\citep[\emph{cf.}][241]{hoefer00}.  I wish to make five
observations about this substantivalist account of a lone hand's
determine handedness.

First, as Hoefer is keen to stress, such an account is only open
to the substantivalist who believes that there are
\emph{primitive} facts about which points of space or spacetime in
two different possible worlds count as the ``same'' point. In
terms that will be more familiar to philosophers, it is not enough
that one be a substantivalist; to give such an account, one must
also be a \emph{haecceitist}.  Since the issue was brought into
focus by Earman and Norton's version of Einstein's ``hole
argument'', many philosophers have concluded that commitment to
such primitive identities, and the corresponding haecceitistic
differences between possible worlds, is not an obvious concomitant
of a belief in the fundamental reality of space or spacetime
\citep[see,
especially,][]{brighouse94,rynasiewicz94,hoefer96}.\footnote{I
should also mention that not all philosophers agree. Belot and
Earman, for example, argue against substantivalists who reject
haecceitism, whom they brand ``sophisticated substantivalists''
\citep{belotearman00,belotearman01}.  For a response to their
arguments, see \citet[Chap.~9]{pooley02}.}

Second, the account only works for possible worlds the
space(time)s of which have the same global topology as that of the
actual world. This is because it is not clear what transworld
identity relations, primitive or otherwise, could hold between two
non-diffeomorphic spaces.

In fact, that the two spaces have the same global topology is not
even sufficient. Let us assume substantivalism and primitive
transworld identity for the sake of argument. There will
nevertheless be spacetime points that are the location of some
instantaneous stage of my left hand in this world but that form a
perfect sphere in some other possible world. All that is required
is that the region they constitute is topologically identical to
my left hand.  The handedness of the hands of this possible world
will thus be undetermined for they will all be equally
(in)congruent to the space actually occupied by my left
hand.\footnote{the restriction to points underlying an
\emph{instantaneous} stage of \emph{my} left hand is incidental.
There are possible worlds in which the spacetime points forming
the worldtubes of every actual hand form worldtubes of perfect
spheres, or of objects whose handedness changes over time etc.}

Third, although \emph{these} complications do not tell against
Hoefer's reconstruction being faithful to Kant's thinking---Kant
and his contemporaries implicitly assumed that the spaces of all
possible worlds were isometric to $E^3$---the reconstruction
certainly does not do justice to Kant's assertion that having a
particular handedness is a matter of having the correct relation
to space \emph{as a unity}. According to Hoefer's Kant, it
consists in having the right relation to \emph{particular} regions
of space, for example, the region which is actually the location
of my left hand. This is something that Kant denies
\citep[365]{kant68}.\footnote{Hoefer does point out that ``no
particular points, lines, rays or regions [of space] are the ones
that have to be mentioned'' \citeyearpar[243]{hoefer00}.  However,
this hardly makes it the case that being of a particular
handedness is a matter of a hand's relation to space as a
\emph{unity}, rather than, say, to space as a \emph{plurality}.
Kant is explicit in his denial that handedness involves a hand's
relation to places (and hence, presumably, to sets of these).}

Fourth, it is evident that the account is surprisingly close to
the relationalist account of handedness just given. In particular,
note how the account `explains' what it is to be a left-handed
hand-shaped region of space. This is held to be merely a matter of
congruence to the actual material hands that we in fact,
\emph{actually}, call ``left''. If one believes in primitive
identity, one can exploit the fact that a particular hand-shaped
region of space exists in a large class of possible worlds to
secure the handedness of material hands in all these world. One is
effectively securing a \emph{vicarious} congruence between
material hands in two different possible worlds by way of
particular hand-shaped regions of space that are supposed to exist
in both. If this is all there is to the substantivalist's
explanation of the handedness of the hands in one-hand worlds,
then the relationalist's assertion that such hands do not have a
determinate handedness starts to look decidedly less
exceptionable.

The final, and related, observation is that the account surely
does not connect with our epistemological situation. We certainly
do not recognize hands as left in virtue of recognizing their
congruence to particular regions of space.  Such regions are
invisible. This underlines the fact that the account is
effectively a marriage of a relational account of handedness with
substantivalism and haecceitism, so as to secure the determinate
handedness of hands in other possible worlds.

\section{The challenge of parity violation}

The conclusion of the first part of this paper is that objects of
opposite handedness that are otherwise identical, such as
idealized left and right hands, do not differ intrinsically in any
way and, furthermore, that their opposite handedness is a matter
of their external spatial relations to each other (and to a
language using community that has assigned quite arbitrary labels
to the two incongruent classes of such objects).  Recently,
philosophers have realized that modern physics appears to suggest
that there is a substantial difficulty with this view.  The
problem goes beyond the mere existence of incongruent
counterparts; it arises from \emph{parity violation}, the fact
that the laws of nature appear to distinguish between left and
right.

The transformation of \emph{parity inversion} is spatial
reflection through the origin.  In the context of quantum field
theory, it is closely connected to two other discrete
transformations, namely \emph{time reversal} (temporal reflection
through the origin) and \emph{charge conjugation} (the interchange
of matter and anti-matter).  If space has an odd number of
dimensions (as the space of our world has) parity inversion maps a
handed object onto an incongruent counterpart.\footnote{Consider
the Cartesian coordinates of a left hand relative to an arbitrary
(left- or right-handed) set of axes. Now consider a passive parity
inversion: a point that originally had coordinates $(x,y,z)$ is
assigned new coordinates $(-x,-y,-z)$. The coordinates of the left
hand with respect to the new coordinate system are the coordinates
of a possible \emph{right hand} with respect to the original
coordinate system.} If the parity transformation is a symmetry of
a theory, it will always map physically possible states of affairs
onto physically possible states of affairs.  Hence if a particular
handed object or process is physically possible, then parity
conservation implies that its incongruent counterpart will also be
physically possible. Conversely, if parity inversion fails to be a
symmetry, then there will be at least \emph{some} cases where it
maps a physically possible state of affairs onto one that the law
prohibits.  If we assume that spatial translations and rotations
\emph{are} symmetries of the theory, then these will be cases
where there is a type of enantiomorphic object or process of one
handedness that is physically possible and yet its incongruent
counterpart is not.\footnote{Actually, this is what is required of
a \emph{deterministic} parity-violating law. If the law is
essentially probabilistic, then parity will also fail to be a
symmetry if \emph{different} probabilities are assigned to a pair
of counterpart yet incongruent processes.}

John Earman \citeyearpar[Chap.~7]{earman89} was the first to
suggest that, while the relational account outlined above may be
able to deal with incongruent counterparts, the fact that a law of
nature violates parity poses a more recalcitrant problem. In fact,
he sees parity violation as having implications for
substantivalist--relationalist debate in much the way Kant
initially thought that incongruent counterparts had. The reason is
that he believes the substantivalist \emph{can} ground the
left--right asymmetry exhibited in processes governed by
parity-violating laws whereas the relationalist cannot.  I shall
shortly question this assumption.

Earman's example of a process that exemplifies such a law involves
the decay of neutral hyperons that was experimentally investigated
by \citet{crawford57} as a test for parity violation.  An example
that may be more familiar is the $\beta$-decay of radioactive
cobalt atoms, the subject of the first experimental confirmation
of parity violation \citep{wu57}. In such a decay the electron and
its antineutrino are preferentially emitted along the axis of
nuclear spin. Given this, there are two, mirror-image
possibilities, depicted in Figure~\ref{fig:co60}. In (a) the
electron is emitted in the same direction as the spin of the
cobalt nucleus, in (b) the electron is emitted in the opposite
direction.

\begin{figure}[h]
\begin{center}
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\begin{picture}(72.67,44.67)
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\put(18.67,27.67){\makebox(0,0)[lb]{Co$^{60}$}}
\put(18.33,38.67){\makebox(0,0)[lb]{e$^{-}$}}
\put(18.00,3.33){\makebox(0,0)[lb]{$\bar{\nu} _{e}$}}
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\put(66.00,27.67){\makebox(0,0)[lb]{Co$^{60}$}}
\put(66.00,38.67){\makebox(0,0)[lb]{e$^{-}$}}
\put(66.00,3.33){\makebox(0,0)[lb]{$\bar{\nu} _{e}$}}
\put(2.33,36.67){\makebox(0,0)[lb]{{\bf (a)}}}
\put(47.00,36.67){\makebox(0,0)[lb]{{\bf (b)}}}
\end{picture}
\end{center}
\caption{$\text{Co}^{60} \rightarrow \text{Ni}^{60} + e^- +
\bar{\nu}_e $} \label{fig:co60}
\end{figure}

The weak interaction, which governs this decay process, fails to
be symmetric under parity inversion.  The decay (a), it turns out,
is much more probable than (b).  In Wu's experiment, a sample of
cobalt 60 was cooled to near absolute zero and then subjected to a
magnetic field to align the nuclear spins: many more electrons
were detected emerging in the direction of nuclear spin than in
the opposite direction.

In terms of this example, here is how Earman puts the challenge to
the relational account of handedness defended above:
\begin{quote}
The failure of mirror image reflection to be a symmetry of laws of
nature is an embarrassment for the relationist account
sketched\ldots for as it stands that account does not have the
analytical resources for expressing the law-like asymmetry for the
analogue of Kant's hand standing alone. Putting some 20th century
words into Kant's mouth, let it be imagined that the first created
process is a [$\text{Co}^{60} \rightarrow \text{Ni}^{60} + e^- +
\bar{\nu}_e $] decay.  The absolutist has no problem in writing
laws in which [(a)] is more probable than [(b)], but the
relationist\ldots certainly does since for him [(a)] and [(b)] are
supposed to be merely different modes of presentation of the same
relational model.  Evidently, to accommodate the new physics,
relational models must be more variegated than initially thought.
\citeyearpar[148]{earman89}
\end{quote}

Without doubt, Earman has put his finger on something.  But one
might wonder whether that the full scale of the problem has been
stated. Two things are worth saying immediately. First, given the
conclusion of the first part of this paper, is it \emph{obvious}
that the absolutist (a.k.a.\ the substantivalist) has no problem
``writing laws in which (a) is more probable than (b)''?

Modulo the qualifications made above, Hoefer's haecceitist
substantivalist can secure the handedness of lone hands in
otherwise empty possible worlds.  In particular, Hoefer's
substantivalist can claim that the reality of space grounds the
genuine distinctness of a world in which the first created process
perfectly resembles (a) and a world in which it perfectly
resembles (b).  However, he does so by claiming that processes of
type (a) stand in different relations to \emph{particular} bits of
space to those in which processes of type (b) stand.  It seems
doubtful that such a substantivalist will want to write relations
to particular bits of space into the laws.  As Hoefer says: ``It
seems wrong for a law of nature to contain reference to a
particular, contingent physical object. But it seems (to me) at
least as wrong for a law of nature to contain reference to
particular bits of space\ldots '' \citep[253]{hoefer00}.

Second, Earman's way of setting up the challenge, in terms of the
``first created process'', suggests a relationalist response that
echoes Herman Weyl's response to Kant's argument.  Weyl wrote,
``Had God, rather than making first a left hand and then a right
hand, started with a right hand and then formed another right
hand, he would have changed the plan of the universe \emph{not in
the first but in the second act}, by bringing forth a hand which
was equally rather than oppositely oriented to the first created
specimen'' \citep[21]{weyl52}.  Similarly, perhaps the
relationalist can maintain that whether the first created process
is a typical decay governed by the weak interaction, or whether it
is a possible but atypical decay, will depend on its incongruence
or otherwise to the majority of subsequent similar decays.  This,
I think, is ultimately what the relationalist has to say. One aim
of the rest of this paper is to highlight some of the costs
involved.

The Weyl-style relationalist is obviously allowed the relationally
acceptable distinction between a world where the first decay
process is typical (i.e., congruent to the majority of subsequent
decays) and one where it is atypical.  He is also allowed the
distinction between (parity violating) worlds where the majority
of decays are handed in the same way and (parity symmetric) worlds
in which decays of opposite handedness occur with equal frequency.
The fundamental problem faced by any account of handedness
according to which incongruent counterparts do not differ
intrinsically is, in those worlds where the majority of decays are
handed in the same way, how can this asymmetry be
\emph{explained}?  If the decay modes (a) and (b) are
intrinsically identical, what could \emph{ground} their different
likelihoods?

The challenge posed by parity violation is thus well-put by
Van~Cleve, who anticipates Hoefer's unease with laws that make
reference to particulars:
\begin{quotation}
God could no doubt \emph{see to it} that certain kinds of
particles always decay into configurations of the same handedness.
But we need to be able to suppose that the result in question
comes about through law rather than divine supervision. How can it
be law that particles always\ldots\ display decay modes of one
orientation rather than another, if orientation is not intrinsic?
If one particle has decayed in left-handed fashion, how does the
next particle `know' that it should do likewise? It's instruction
cannot be to trace a pattern of a certain intrinsic description;
it can only be to do what the first particle did.

The problem here is not `action at a distance', though perhaps
that will trouble some. It is rather that the required laws would
make ineliminable reference to particular things, whereas it is
generally supposed to be of the essence of laws that they state
relations of kind to kind. \citep[21--2]{vancleve91}
\end{quotation}

In a moment I shall suggest that in one respect Van~Cleve is
wrong; the problem \emph{is} action at a distance and not
ineliminable reference to particular things.  But to see how
reference to particular things---whether they be particle decays
or regions of space---can be avoided, we need to review some of
the details of the law that describes parity-violating processes.

\section{A relational account of parity violation}

That Nature treats left- and right-handed varieties of certain
processes differently is puzzling.  And yet we have an extremely
well-confirmed physical theory describing how it does so.  How
does the mathematics of the theory work?  Does it do so by
revealing that left-handed and right-handed varieties of handed
objects differ intrinsically after all?

The most fundamental description of parity-violating interactions
so far formulated is that given by the Weinberg-Salam gauge field
theory, part of the Standard Model.\footnote{I review some
relevant details of the Weinberg-Salam theory in an appendix.}
The Weinberg-Salam theory treats elementary particles, such as
electrons and quarks, as excitations of Dirac quantum fields
interacting via gauge boson fields (the photon, the $W^{+}$ and
$W^{-}$, and the $Z$). Parity violation results from the fact that
the theory treats the `left' and `right' chiral components of the
same Dirac field quite differently; they couple to the interaction
fields in different ways. In particular, `right-handed' particle
fields do not couple to the $W$ bosons at all.

So what does it mean to call a component of a field `left handed'?
The Dirac field can be thought of as the sum of two component
fields, the left and right chiral components, which, in the zero
mass limit, correspond to particles of definite and opposite
\emph{helicity}. The helicity of a particle is the projection of
its spin in its direction of motion. Helicity eigenstates of a
spin-$\frac{1}{2}$ particle involve the spin being either aligned
or anti-aligned with the particle's direction of motion. These
constitute two incongruent, `handed' objects. By definition,
\emph{left-handed} massless particles are particles of negative
helicity (their spin is opposite to their direction of motion),
while \emph{right-handed} particles are particles of positive
helicity.

\begin{figure}
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\unitlength 1mm \linethickness{0.4pt}

\begin{picture}(72.67,24.67)

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\put(18.67,7.67){\makebox(0,0)[lb]{$\psi_{\mathrm{R}}$}}
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\put(66.00,7.67){\makebox(0,0)[lb]{$\psi_{\mathrm{L}}$}}
\put(66.00,18.67){\makebox(0,0)[lb]{$c$}}

\end{picture}
\end{center}
\caption{chirality in the zero mass limit}
\end{figure}

Helicity and chirality are not quite the same thing, however. A
helicity eigenstate of a massive particle will involve both left-
and right-handed pieces. Moreover, while the chirality of a
particle is Lorentz invariant, the helicity of a massive particle
is not. For example, one can Lorentz boost by a large enough
velocity in the direction of the particle's motion so as to
reverse that direction of motion while leaving the direction of
spin unchanged. This cannot be done, of course, if the particle is
massless and thus travelling at the speed of light. The helicity
of a \emph{massless} particle \emph{is} an invariant property
under the (restricted) Lorentz group.

A spinning object defines an axis: that about which it is
spinning. For a given axis there are then two possibilities
involved: if one looks along the axis of spin from a given
direction, the object will either appear to be spinning clockwise
or anticlockwise. These two possibilities are represented by
associating each with a direction: the spin vector points along
the axis of spin away from the point of view from which the
spinning appears clockwise. This is equivalent to the definition
of angular momentum vector $\mathbf{l}$ as $l^{i}=\varepsilon
^{ijk}x^{j}p^{k}$ where $\varepsilon ^{ijk}$ is completely
antisymmetric in its three indices, $\varepsilon ^{123}=+1$ and
the components of $\mathbf{x}$ and $\mathbf{p}$ and given with
respect to a conventional, \emph{right-handed} set of Cartesian
axes.

In the last analysis, in each case, the convention for associating
a direction with a spinning object can only be specified via
ostension.  We can explain what we mean by left-handed and
right-handed particles in terms of their relations to right-handed
sets of axes, or in terms of their relations to typical clocks.
But, if Bennett's Kantian hypothesis is correct, the meanings of
these terms cannot be conveyed without ostension.

Turning now to the case of negative and positive helicity
particles, one sees from the conventionality of the definition of
the direction of spin, and from the need for ostension in
specifying this direction, that to explain the difference between
negative and positive helicity one must also ultimately appeal to
ostension. The parallel with left and right hands is obvious. An
\emph{intrinsic} description true of a negative helicity particle
will also be true of a positive helicity particle. Yet we can
understand why there are \emph{two}, incongruent, types in purely
relational terms (i.e., in terms that do not presuppose an
intrinsic difference). For two spin-$\frac{1}{2}$ particles of
definite helicity travelling in the same direction they can be
spinning in the same sense, or the opposite sense.

It turns out that the mathematics of parity violation does not
involve treating being left-handed and being right-handed as
different substantive and intrinsic properties.  As is explained
in the appendix, left and right components of the fields are
distinguished in terms of their differing congruence relations to
the right-handed coordinate systems with respect to which the
theory is standardly written. One then goes on simply to assert of
these two components that they interact differently.

This difference in the way they interact is not further explained
in terms of different intrinsic properties possessed by the
particles.  Although I note in the appendix that the left and
right components of the field are assigned different values of
``weak hypercharge'' and ``weak isospin'', these are not
properties that \emph{explain} their particular couplings to the
gauge fields. For two varieties of particle to have the particular
values of weak hypercharge that they have, for example, \emph{just
is} for them to couple to the gauge field $B_\mu$ with the
relative strengths that they do.  In fact, it seems plausible
that, quite generally, particle varieties are only individuated in
terms of their particular place in a network of differently
interacting particles.

Moreover, it cannot be the helicity of a particle alone that
determines which type of interactions it can undergo.  The reason
is that the `left-handed' Dirac field component
$\psi_{L}(x)$---and its adjoint, $\overline{\psi_{L}}(x)$---are
associated with left-handed particles and \emph{right-handed}
antiparticles. (In the massless limit these correspond to negative
and positive helicity eigenstates respectively.)  Therefore the
theory not only violates parity $P$, it also violates \emph{charge
conjugation} $C$, the interchange of particles with their
antiparticles. However, the fragment of the theory described in
the appendix is invariant under the \emph{combined} transformation
$CP$. Just as only left-handed electrons and quarks couple to the
$W$s, only right-handed positrons and anti-quarks couple to
them.\footnote{The full Standard Model, involving all three
generations of quarks, violates $PC$ symmetry.  However, any
theory based on a Lorentz-invariant Lagrangian $\mathcal{L}(x)$
formed from products of quantum fields at the point $x$ will be
invariant under the combined transformation $CPT$.} So a
description of the left-handed $\beta$-decay of a cobalt atom in
intrinsic, relational terms will be equally true of a right-handed
decay of an antimatter cobalt nucleus.

Does this fact, by itself, vindicate a relational account of
handedness?  Simon Saunders has suggested that it does
\citep{saunders00a}. His point is that the parity-violating law
does not, after all, `pick out' a particular handedness. Anything
that is possible for a particle of one handedness is possible for
particles of the opposite handedness, although it \emph{may} only
be possible for oppositely handed particles if they are also
antiparticles of the first.

Just as the relationalist seeks to identify putatively distinct
possible worlds containing nothing but single hands that
supposedly differ solely in the sense of their handedness,
Saunders urges that we should treat models of a parity-violating
but $CP$-symmetric theory that are related by a global
$CP$-trans\-for\-mation as different representations of the
\emph{same} state of affairs.  If we describe a world in which the
the first created process is a decay of a cobalt atom, there is no
fact of the matter whether it was a decay of a matter nucleus
emitting a right-handed antineutrino, or the decay of an
antimatter nucleus emitting a left-handed neutrino.  (Of course,
if \emph{per impossibile} the first created process of the actual
world was a cobalt decay there would be a fact of the matter.  But
this would be a \emph{relational} fact: did the decay involve the
emission of a neutrino of the same handedness as particles that we
in fact call ``right-handed'' and a particle of the same charge as
particles that we in fact call ``electrons''; or did it involve
the emission of a neutrino of the same handedness as particles
that we in fact call ``left-handed'' and a particle of the same
charge as particles that we in fact call ``positrons''?)

Unfortunately, $CP$ (or even $CPT$) symmetry does not by itself
save the relationalist.  Nor, indeed, is it even a necessary
component of the relationalist's account.  To see this, consider
the following toy models. The first involves a possible world
whose fundamental objects are hand-shaped and come in two
varieties: ``red'' and ``green''. Red hands are never created, but
they can ``decay'' into green hands. Now let us suppose that only
red `left' hands can decay into green `left' hands. Red `right'
hands never decay and no green `right' hands exist at all. This
decay law clearly violates parity.

We can, however, extend our example so that it involves a
$P$-violating but `$CP$' symmetric law. In this second possible
world, we now imagine that the red hands are `charged' in that
they attract or repel one another: similarly charged hands repel
each other whereas oppositely charged hands attract each other.
Now both left and right red hands can be `negatively' and
`positively' charged. However, suppose that it is the case that
only `negatively' charged red left hands can decay into green
(left) hands whereas `positively' charged right hands can decay
into green right hands.  The law is now $CP$ symmetric: if a
particular hand can decay, then so can its incongruent,
charge-reversed counterpart.

Is this $CP$-symmetric law really more susceptible of a
relationalist interpretation than the first, $P$-violating law?
The problem that the relationalist still faces is to explain why
negatively charged red right hands cannot decay into similarly
oriented green hands when negatively charged red left hands (i.e.,
hands that are identical apart from the sense of their handedness)
can. Relationalism denies that there is some intrinsic difference
between the two types of hand that can ground and explain their
different interactions. The fact that \emph{positively} charged
red right hands can decay into similarly oriented green hands does
nothing to ameliorate the problem.  Similarly, the relationalist
is at a loss to explain why only left-handed \emph{electrons}
couple to $W$ bosons.  How can it be that left-handed and
right-handed electrons interact differently if the relationalist
is correct in his deflationary account of what their being left-
or right-handed consists in?  The fact that right-handed
\emph{positrons} can couple to $W$ bosons is of little comfort.

There is another reason for being wary of invoking $CP$ (or $CPT$)
symmetry to save a relational account of handedness: it fails to
secure for the relationalist all that he desires. In his
correspondence with Clarke, Leibniz, the arch-relationalist,
insisted that two putatively distinct possible worlds differing
solely over where the material universe was located in space were
really just two ways of differently describing a single
possibility.  It is the homogeneity of space---that translations
are a \emph{symmetry} of Newtonian mechanics---that means that if
one of these worlds is \emph{physically} possible, then so is the
other.  Similarly, the relationalist will wish to see two models
of a $P$-violating but $CP$-symmetric theory that are related by a
global $CP$ transformation as but two ways of representing a
single state of affairs.

What, though, should they say about the model one obtains from one
model from such a pair by performing a global parity
transformation \emph{without} also interchanging matter with
antimatter? In the case of our second toy model, we obtain a world
where it is negatively charged \emph{right} hands and positively
charged \emph{left} hands that can decay.  In the case of the a
model of the Standard Model, we describe a world where
\emph{right-handed electrons} and \emph{left-handed positrons}
couple to $W$ bosons. The relationalist who sees the existence of
a symmetry as a prerequisite of being able to identify the
possibilities represented by models related by a non-trivial,
global transformation must deny that these situations are
equivalent to the original ones.  He must even deny that they obey
the same laws!

Something surely has gone wrong here. In what way does the
parity-imaged model differ from the original? We call the quarks
that couple to the $W$s in the first model ``left-handed'', and
call the quarks which couple to the $W$s in the second model
``right-handed'', but what is the difference between them? In
terms of the functional roles they play within the models, they
are indistinguishable. If the original model was detailed enough
to include human experimenters referring to left-handed quarks,
then their counterparts in the second model will call the
supposedly right-handed quarks ``left-handed'' too.  The
relationalist intuition is surely that the left-handed quarks of
the first model should be \emph{identified} with the so-called
right-handed quarks of the second.

Despite the fact that the worlds described by the two models
display a law-like asymmetry between left and right, and despite
the fact that the models are \emph{nominally} the mirror images of
each other, they should be regarded as solutions of a \emph{single
theory}, and the two models should be judged to describe a single
possibility.

So let us now return to Earman's claim that the relationalist
lacks the ``analytical resources'' to describe a law that can
embrace both models.  The relationalist must eschew terms such as
``left'' or ``right'', and he must not rely on a formulation that
makes \emph{implicit} use of such terms by relating the physics
to, for example, right-handed coordinate systems.  It seems,
however, that the relationalist can indeed provide such a law.

Consider the first toy model again.  The claim that all red hands
which decay into green ones are \emph{handed in the same way}
embraces both the original possible world and its supposedly
distinct parity-image.  What is there to stop the relationalist
claiming that this is a \emph{lawlike} statement and, moreover,
that it is primitive: it cannot be further explained in terms of
more fundamental laws? Turning to the second, $CP$-symmetric, toy
model, we can state that red hands that \emph{repel each other}
and which can decay into green hands are handed in the same way;
red hands that attract each other and which can decay into green
hands are handed in the opposite way.

Saunders offers a similar schematic law for $\beta$-decay:

\begin{description}
 \item [PC] Charge-conjugate $\beta$-decay processes are
oppositely oriented.
 \end{description}

He stresses that this a $PC$ invariant statement of $P$ violation.
What is important for the present discussion is that it is
\emph{also} a $P$ invariant statement, even though it is an
expression of $P$ violation. The statement is true both of a model
where ``left-handed'' electrons couple to $W$ bosons \emph{and} of
a model where ``right-handed'' electrons do so. Of course, the law
needs to be extended to include all varieties of fields---in
particular, it must say something positive about the interactions
which the electron-positron pair that cannot be produced in
$\beta$-decays can undergo---but the outlines of how this is to be
done are clear enough.

\section{Orientation Fields}

In characterizing the relationalist's position in this way, I am
very close to Carl Hoefer.  He writes: ``The correct perspective,
for either relationists or substantivalists, is this: P-violating
laws mandate \emph{that there shall be a certain, qualitative,
spatial asymmetry} in events.  They do not explain the asymmetry
or how it arises\ldots Bringing in enantiomorphic objects allows
one to `anchor' the asymmetry descriptively, but is in no way
explanatory of the asymmetry, nor do such objects become `referred
to' in the laws by being so used'' \citep[253]{hoefer00}.  In the
case of the Standard Model, the enantiomorphic objects in question
are actual left and right hands that are linked to the law, as we
have seen, via the conventions that define what we mean by
\emph{right-handed} coordinate systems.

However, Hoefer also claims that parity violating laws are
``purely phenomenological'' and that this should, at least in
part, ease any worries that we might have over the fact that the
law-like asymmetry in phenomena is not ultimately explained.  This
seems to me to be contestable.  The Standard Model is not a
``purely'' phenomenological law.  There is a world of difference
between the early descriptions of the weak interactions in terms
of $V-A$ currents and the Standard Model together with the
understanding that it provides us with of these truly
phenomenological laws. The asymmetry in the phenomena \emph{is}
explained, albeit only in terms of a deeper asymmetry that is not.
We postulate fundamentally handed particles (corresponding to
massless spin-$\frac{1}{2}$ Dirac fields) and attribute different
sets of interactions to oppositely handed fields. \emph{Why}
oppositely handed fields of the same particle type differ in this
way is not explained.  But there is no reason why the fact that
they do should be seen as a phenomenological, rather than as a
fundamental, fact.

Lack of a further explanation should not \emph{per se} be seen as
a problem for the relationalist.  However, there is one feature of
the relationalist's story that some will object to, and it is a
feature that can lead back to substantivalism.

Recall that Van~Cleve alleged that the problem faced by the
relationalist was ineliminable reference to particulars rather
than ``action at a distance''.  We have seen that there is no
ineliminable reference to particulars.  However, the basic form of
our proposed law---that all objects or processes of a certain
relationally specifiable type are \emph{handed in the same
way}---is, in a certain specific sense, highly \emph{non-local}.
For the relationalist, being handed in the same way just is for
two things to stand in certain spatio-temporal, and quite possibly
spacelike, relations.

It is time to consider the recent `reconstruction' of Earman's
argument offered by Nick Huggett \citep{huggett00}.  One of
Huggett's central claims is that rather specific geometrical
structures are involved in a proper formulation of
parity-violating laws. He illustrates this claim with his own toy
model quantum theory, involving two particles in one dimension,
coupled by the following asymmetric potential:
\begin{equation}
V(x_1,x_2) = \lambda(x_1 - x_2) + \mu (x_1 - x_2)^2.
\end{equation}
In this theory the two directions in its 1-dimensional space are
not on a par.  On measurement of their positions, the probability
of finding particle 1 `to the left of' particle 2 (i.e., with a
more negative position coordinate, relative to the coordinate
expression of the potential given above) is greater than finding
them the other way around.\footnote{Note that one unfortunate
aspect of Huggett's example can be ignored.  His theory violates
both parity symmetry \emph{and permutation symmetry}.  It is not
that one is more likely to find a particle of one \emph{type} to
the left of a particle of another type; rather one is more likely
to find particle 1, \emph{that particular particle}, to the left
of particle 2.  The fact remains that it is solely the theory's
violation of parity that requires the introduction of an
orientation field when expressed in a coordinate-free way.
Similarly the coordinate-free expression of the Standard Model
requires the introduction of an orientation field, even though
this theory does not violate permutation symmetry.}

Huggett's claim is that the theory is not well-defined until an
`arrow of space' has been given, enabling us to say whether $x_1 <
x_2$ or $x_2 < x_1$ `in absolute terms, not just relative to some
arbitrary coordinates' \citep[233]{huggett00}.  He links this
claim to the fact that the coordinate-free expression of the
potential will involve the explicit introduction of an
\emph{orientation field} (in the case of the 1-dimensional theory,
this field is simply a normalized 1-form).  Similarly, if we were
to express the equations of the Standard Model in a
coordinate-free way, we would need to introduce an orientation
field explicitly, and distinguish left- and right-handed
components of fields by their relations to it, rather than to a
standard coordinate system.

Huggett then puts the argument for substantivalism as follows,
deliberately paraphrasing Earman's version of why the postulation
of inertial structure to ground the distinction between absolute
and relative motion licences the move to substantivalism:
\begin{quote}
\ldots the `absolutist' asserts that ``the scientific treatment of
motion \ldots requires some absolute quantities \ldots such as
handedness.  To make these quantities meaningful requires the use
of an orientation, and this structure must be a property of or
inhere in something distinct from bodies.  \emph{The only
plausible candidate for the role of supporting the nonrelational
structures is the spacetime manifold}.''
\citetext{\citealp[236]{huggett00}; \emph{cf}.\
\citealp[125]{earman89}}
\end{quote}

The parallel drawn here is suggestive and worth pursuing.  Pure
inertial motion can be thought of as manifesting rather noteworthy
non-local correlations.  Assuming the correctness of Newtonian
mechanics, one can, from the relative motions of just three
force-free bodies, construct a spatio-temporal coordinate system
with respect to which all three of these bodies are moving
uniformly and in straight lines.\footnote{An elegant demonstration
of this fact was given by Tait \citep{tait83}.  For a nice account
of it, see \citet[Chap.~6]{barbour99a}.}  This is already a highly
non-trivial fact that one might feel calls for explanation, given
that, as force-free bodies, the three bodies are supposed to be
moving quite independently of each other.  What is perhaps more
striking is that every other force-free body is also moving
inertially with respect to the coordinate system defined by the
first three.  The substantivalist offers a \emph{local}
explanation of these non-local correlations. According to him, the
laws of motion constrain the motions of such bodies at each point
of spacetime to be geodesics, as defined at each point by the
affine connection.

Similarly, by reifying the orientation field, we can offer a local
explanation of the non-local correlations between $\beta$-decays:
that all neutrinos emitted in such decays are handed in the same
way. According to Huggett's substantivalist, these correlations
follow from the fact that the laws postulate that, at each point
of spacetime, only quarks standing in one of the two possible
relations to the orientation field \emph{at that point} can couple
to $W$ bosons.

\emph{If} the introduction of a real orientation field provides a
genuine and local explanation of the congruence of all
$\beta$-decays at no cost we should surely admit such a field into
our ontology.  The question is, of course, whether the explanation
is genuine, and whether there are costs. One might also wonder
whether such a field supports spacetime substantivalism: does such
a field obviously represent \emph{spacetime structure} (as does
the metric field of relativistic theories\footnote{This claim is,
perhaps, controversial. Here I am siding with, e.g.,
\citet[318]{maudlin89} and \citet[459--60]{hoefer98} against
\citet[193--4]{rovelli97}. For further discussion, see
\citet[Chap.~4]{pooley02}.}) rather than just another real,
physical field in spacetime?

Hoefer has objected to the move on different grounds.  He holds
that it amounts to nothing more than writing reference to
particular bits of space into the laws:
\begin{quote}
It seems wrong for a law of nature to contain reference to a
particular, contingent physical object. But it seems (to me) at
least as wrong for a law of nature to contain reference to
particular bits of space\ldots  That this is what is going on may
be masked by talk of ``absolute structures'' or ``a preferred
$n$-form defined at all points'', or something of this nature. But
such terminology, while not literally incorrect, really only
disguises the dependence on primitive identity to make the
distinctions between orientations for us. \citep[253]{hoefer00}
\end{quote}

The reader might be surprised at this assertion.  Where, in the
foregoing discussion of orientation fields, was reference made to
particular bits of space?  The reification of an orientation
fields does not entail a commitment to primitive identity. Care is
needed, however, as is illustrated by the following quite from
Huggett, which follows his explicit introduction of an orientation
field into his toy-model theory:
\begin{quote}
At this point it is worth noting for clarity that there is also a
conventional aspect to such handed theories.  For suppose the
arrow of space now runs in the opposite sense; if $V_A$ remains
the potential, then it will have the opposite handedness in space
(compared to the original, or compared to some external bodies)
and the system will behave differently.  But if the potential also
changes, $V_A \rightarrow -V_A$, then of course the dynamics will
be as before.  Thus, it does not make sense in this situation to
ask in which direction the arrow of space runs, independently of a
given Hamiltonian, and likewise it makes no sense to ask which
sign of $V_A$ is correct, independently of an arbitrary choice of
arrow.  Thus the two possible arrows and two possible Hamiltonians
only allow two distinct theories not four.  This point
acknowledged, we can talk of \emph{the} arrow and \emph{the}
Hamiltonian and bear in mind the freedom this actually leaves.
\citep[234--5]{huggett00}
\end{quote}

I agree with Huggett that there are certainly no more than two
possible theories, not four.  However, two readings of this
passage are possible.  And read in one way, precisely the wrong
identifications are being advocated.

The orientation field is either supposed to be a real, physical
field, or is supposed to represent some genuinely asymmetric
structure of space or spacetime itself.  If this is the case, then
one \emph{cannot} identify a theory that assigns a certain
probability to the vector from particle 1 to particle 2 being
aligned with the arrow defined by the orientation field, with a
theory that assigns precisely that probability to the case where
the two vectors are in the \emph{opposite} alignment. Similarly, a
theory that asserts that all electrons which are ``congruent'' to
an orientation field couple to $W$ bosons and those which are
``incongruent'' do not, cannot be identified with a theory that
predicts the same phenomena by asserting that all electrons which
are \emph{in}congruent to an orientation field couple to $W$
bosons.

However, the anti-haecceitist substantivalist who follows Hoefer
in denying primitive identity relations between the spacetime
points of different possible worlds will not be able to
distinguish worlds that involve the same relations between
orientation field and the matter fields, but that differ solely in
terms of the relations that all of these fields bear to particular
points of spacetime. But this is exactly the distinction that
Huggett might appear to be upholding when he talks, for example,
of the ``arrow of space running in the opposite sense.''  It seems
that we are here being asked explicitly to imagine the orientation
field bearing a different relationship to particular points of
space.\footnote{The alternative reading of Huggett's passage
involves no such commitment to haecceitism.  Instead one is merely
noting that, \emph{relative to a fixed coordinate system}, one can
represent the asymmetric structure attributed to spacetime in two
equal good ways.  Although the coordinate system is kept fixed,
one is contemplating a \emph{passive} transformation that results
in the orientation field of the mathematical model bearing a
different relation to the physical orientation field (or to the
asymmetric spacetime structure).  Huggett has indicated that it
was this freedom he was intending highlight (private
communication).}

So we can introduce an orientation field to ground a local
explanation of the non-local asymmetries that the relationalist
must postulate as brute, lawlike facts, in a way that doed not
involve an implicit commitment to haecceitism and primitive
identities.  Nevertheless, it does appear to involve an
unavoidable commitment to the reality of differences that are
unobservable in principle: the theory that has only electrons
``congruent'' to such a field coupling to $W$ bosons and the
theory that has only electrons ``incongruent'' to such a field
coupling to $W$ bosons must be regarded as distinct theories, even
though they are observationally indistinguishable. This problem is
not merely the result of viewing the orientation field as a
physical field that is distinct from spacetime.  If one were to
insist that it is spacetime itself that has an intrinsically
asymmetric structure, and that the orientation field is just a
mathematical device to encode such structure, one still would not
be able to collapse the four formally distinct theories down to
one.  As Huggett notes (on the intended reading of the above
quote), one is free to chose either of the the two possible
orientation fields as encoding such structure. But this does not
tell against there being a genuine metaphysical distinction
between worlds where electrons bearing one type of relation to
this structure interact in a certain way and worlds in which
electrons bearing the opposite relation interact in an identical
way.  The only way to avoid postulating such a distinction is to
adopt the relationalist's account of parity violation, together
with its brute, lawlike, non-localities.

\section*{Acknowledgements}

I am grateful to Harvey Brown, David Wallace, Mauricio Suarez,
James Ladyman, Leah Henderson, Jeremy Butterfield, Carl Hoefer,
Nick Huggett and Bill Child for comments on material for this
paper and, particularly, to Simon Saunders for many discussions of
handedness and parity violation.  I would like to thank the
editors for the invitation to contribute to this volume and to
thank Katherine Brading especially for helpful and detailed
comments on an earlier draft.  Work for this paper was partly
supported by the Arts and Humanities Research Board of the British
Academy.

\appendix

\section{The weak interaction}

I should start by stressing that there are many reasons for
thinking that the Standard Model is far from a final theory, not
the least of which is the large number of unexplained parameters
involved that must be fixed by hand for the theory to tally with
experiment. There is also, as yet, no consensus on the correct way
to extend the Standard Model so as to include neutrino mass.
Indeed, this caveat is of relevance to our topic because some
suggestions involve theories which are fundamentally parity
symmetric. The standard electro-weak theory results from parity
symmetry being \emph{spontaneously broken} in such theories.

I propose to set these issues to one side.  Treating the
Weinberg-Salam theory as a fundamental theory poses the most
severe challenge for any account of handedness according to which
handedness is an extrinsic, rather than an intrinsic, matter.  So
if such an account can be defended in this context, \emph{a
fortiori} it should be defensible in others.

Central to the Standard Model are the left-handed and right-handed
components of the various Dirac fields.\footnote{In the following,
I have drawn upon \citet{peskinschroeder95} and unpublished
lecture notes by I.T.~Drummond and H.~Osborn.}  Mathematically
they are described in terms of the $\gamma$-matrices.  In
particular, for a Dirac field $\psi$, one has:
\begin{equation}
\psi _{L}=\frac{1}{2}\left( 1-\gamma^{5}\right) \psi ,\quad \psi
_{R}=\frac{1}{2}\left( 1+\gamma^{5}\right) \psi,
\end{equation}
where $\gamma^{5} = \gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. The
$\gamma$-matrices might appear to be defined independently of our
coordinate conventions: all that is required is that they obey the
anti-commutation relations:
\begin{equation} \label{comms}
\{\gamma ^{\mu },\gamma ^{\nu }\}=2g^{\mu \nu }I \text{ ,}
\end{equation}
and $g^{\mu \nu }$ is left--right symmetric. However, they are
tied to the coordinate system through the Dirac equation:
\begin{equation} \label{dirac}
(i\gamma ^{\mu }\partial _{\mu }-m)\psi (x)=0\text{ .}
\end{equation}
If we reverse the sense of our coordinate system, say by
relabelling the $x$- and $y$-axes, we switch the roles of
$\gamma^1$ and $\gamma^2$.  Since, from eqn~\ref{comms}, these
anti-commute, $\gamma^5 \mapsto - \gamma^5$ and so \emph{the
mathematical description of the left-handed component of a
fermionic field with respect to a right-handed set of axes, is
exactly the same description one gives to the right-handed
component with respect to a left-handed set of axes}. The standard
mathematical descriptions of the chiral components of a fermionic
field thus clearly relate them, more or less explicitly, to the
the handedness of the Lorentz chart with respect to which the
physics is formulated.  There is nothing in the standard
mathematical description of a left-handed field that is not
equally suited, given different conventions, to describing a
right-handed field.\footnote{I am indebted to David Wallace for
this way of seeing how the handedness of field components is
defined in terms of the handedness of the coordinate chart.}

Having introduced the left-handed and right-handed components of
the Dirac field, we can see how parity violation is implemented in
the Weinberg-Salam theory.  The theory is based on a Lagrangian
density involving the Dirac lepton fields, the gauge fields and a
scalar field, the Higgs field. The Lagrangian is invariant under
certain local $SU(2)\times U(1)$ gauge transformations. The
standard prescription followed in constructing a gauge field
theory is to start with a Lagrangian that is invariant under some
set of global gauge transformation and then create a locally gauge
invariant theory by replacing the derivative operators in the
original Lagrangian with ``covariant derivatives'' involving
compensating gauge fields to ensure the required invariance. In
this case, the original Lagrangian is the standard
(parity-symmetric) Dirac Lagrangian for \emph{massless} fields
(for simplicity I only consider interactions involving electrons,
positrons and their associated neutrinos):
\begin{equation}
\mathcal{L}_{\text{kin}}(x)=\overline{\psi _{e}}i\gamma .\partial \psi _{e}+%
\overline{\psi _{\nu }}i\gamma .\partial \psi _{\nu }\text{ .}
\end{equation}
No mass terms have been included because to construct a
parity-violating gauge theory, symmetries under which the left and
right chiral components transform \emph{differently} must be
gauged. This prohibits mass terms which mix left and right
components and are thus not invariant under such transformations.

We now rewrite the Lagrangian as:
\begin{equation}
\mathcal{L}(x)=\bar{L}(x)i\gamma .\partial L(x)+\bar{R}(x)i\gamma
.\partial R(x)\text{ .}  \label{ws free lagrangian}
\end{equation}
where $L$ is a \emph{doublet} involving the left-handed components
of the neutrino and electron fields. Writing $\psi _{\nu }(x)=\nu
(x)$ and $\psi _{e}(x)=e(x)$:
\begin{equation}
L(x)=\binom{\nu_{L}(x)}{e_{L}(x)}\text{ .}
\end{equation}
$R$ is a singlet involving the right-handed component of the
electron field: $R(x)=e_{R}(x)$.\footnote{Although I have not
included the right-handed component of the neutrino field, the
fact that neutrinos have mass means that the Standard Model must
be extended to include it.  The simplest way of doing so would be
to include the field as a separate singlet term.  However, as
mentioned above, there are many rival proposals which experiment
has yet to decide between.}

$\mathcal{L}$ is invariant under the $SU(2)$ transformations:
\begin{equation}
L(x) \rightarrow e^{\frac{1}{2}i\mathbf{\alpha .\tau }} L(x),
\quad \bar{L}(x) \rightarrow e^{-\frac{1}{2}i\mathbf{\alpha .\tau
}}\bar{L}(x), \quad R(x) \rightarrow R(x)\text{ ,}
\end{equation}
where $\tau$ are the $2\times 2$ Pauli matrices.  It is also
invariant under independent $U(1)$ phase transformations of $L$
and $R$ separately. In particular, it is invariant under the
separate phase transformations $L\rightarrow
e^{-i\frac{1}{2}\chi}L$ and $R\rightarrow e^{-i\chi }R$ which may
be written $\psi
\rightarrow e^{i\chi Y}\psi $ where $Y= -\frac{1}{2}$ for $L$ and $%
Y= -1$ for $R$. These transformations are taken as the $U(1)$
transformations in an $SU(2)\times U(1)$ global symmetry group in
order that the resulting gauge theory is invariant under local
gauge transformations generated by the electric charge and
electromagnetism is recovered.

A local $SU(2)\times U(1)$ transformation is then written:
\begin{equation}
\psi \rightarrow e^{i\mathbf{\alpha }(x).\mathbf{T}+i\chi
(x)Y}\psi (x)
\end{equation}
where $\mathbf{T}= \frac{1}{2}\mathbf{\tau }$ acting on $L$ and
$\mathbf{T}= \mathbf{0}$ acting on $R$. A gauge invariant
Lagrangian is obtained by replacing the derivatives in eqn~\ref{ws
free lagrangian} with the covariant derivatives
\begin{equation}
D_{\mu }=\partial _{\mu }-ig\mathbf{A}_{\mu
}(x).\mathbf{T}-ig^{\prime }B_{\mu }(x)Y\text{ .}
\end{equation}
The result is:
\begin{equation}
\mathcal{L}=\mathcal{L}_{\text{kin}}+g\bar{L}\gamma ^{\mu }\frac{1}{2}%
\mathbf{\tau }L.\mathbf{A}_{\mu }+g^{\prime
}(\frac{1}{2}\bar{L}\gamma ^{\mu }L+\bar{R}\gamma ^{\mu }R)B_{\mu
}\text{ .}  \label{ws lept gauge}
\end{equation}

The consequences of gauging a symmetry group under which left and
right field components transform differently are now manifest.
Since $R$ is a singlet under the chosen $SU(2)$ transformations,
it does not couple to the three component fields of
$\mathbf{A}_{\mu }$ at all. That the left and right fields are
assigned different values of ``weak hypercharge'' $Y$ means that
the strengths of their coupling to the $U(1)$ gauge field $B_{\mu
}$ is different.

The quantum field theory derived from this Lagrangian together
with the appropriate Lagrangian for the free gauge fields is not
yet empirically adequate because the four gauge fields all
correspond to \emph{massless} gauge bosons. It was known
experimentally that there is only one massless gauge boson
involved in electromagnetic and weak interactions, the photon. The
solution is to exploit the mechanism of ``spontaneous symmetry
breaking''. A scalar field with a gauge invariant potential term
that has a minimum for non-zero values of the field is postulated.
As a result of their coupling to this so-called Higgs field, all
but one of the gauge fields acquire a mass and, with the right
transformation properties under the gauge transformations assigned
to the Higgs field, the massless field corresponds to the $U(1)$
gauge group of electromagnetism. The details need not concern us.
All we need is that the `physical' gauge fields corresponding to
gauge bosons of definite mass are given by
\begin{equation}
\begin{split}
W_{\mu } &= \frac{1}{\sqrt{2}}\left( A_{1\mu }-iA_{2\mu }\right)
\text{ ,}
\\
Z_{\mu } &= \cos \theta _{W}A_{3\mu }-\sin \theta _{W}B_{\mu
}\text{ ,}
\\
A_{\mu } &= \sin \theta _{W}A_{3\mu }-\cos \theta _{W}B_{\mu
}\text{ ,}
\end{split}
\end{equation}
where the Weinberg angle $\theta _{W}$ is defined by $\tan \theta
_{W}=g^{\prime }/g$. Making the identification $e=g\sin \theta
_{W}$, eqn~\ref{ws lept gauge} can be rewritten as:
\begin{equation}
\mathcal{L}=\mathcal{L}_{\text{kin}}+\frac{g}{2\sqrt{2}}(J^{\mu
}W_{\mu }+J^{\mu \dagger }W_{\mu }^{\dagger })+ej_{e.m.}^{\mu
}A_{\mu }+\frac{g}{2\cos \theta _{W}}J_{n}^{\mu }Z_{\mu }\text{ ,}
\label{ws lept phys}
\end{equation}
where the weak, electromagnetic and weak neutral currents are
defined as follows:
\begin{equation}
\begin{split}
J^{\mu }(x) &= \bar{L}(x)\gamma ^{\mu }(\tau _{1}+i\tau _{2})L(x)=\bar{\nu}%
(x)\gamma ^{\mu }(1-\gamma _{5})e(x) \\
j_{e.m.}^{\mu } &= \bar{L}\gamma ^{\mu }\frac{1}{2}(\tau _{3}-1)L-\bar{R}%
\gamma ^{\mu }R=-\bar{e}\gamma ^{\mu }e \\
J_{n}^{\mu } &= \bar{L}\gamma ^{\mu }\left( \cos ^{2}\theta
_{W}\tau
_{3}+\sin ^{2}\theta _{W}1\right) L-2\sin ^{2}\theta _{W}\theta _{W}\bar{R}%
\gamma ^{\mu }R \\
&= \frac{1}{2}\left[ \bar{\nu}\gamma ^{\mu }(1-\gamma _{5})\nu -\bar{e}%
\gamma ^{\mu }(1-\gamma _{5}-4\sin ^{2}\theta _{W})e\right] \text{
.}
\end{split}
\end{equation}
Here again we see that only the left-handed fields couple to the
$W$ bosons and that the coupling strengths of the left- and
right-handed fields to the $Z$ are different. Only the left- and
right-handed couplings of the electron/positron field to the
photon $A$ is symmetric between left and right.

Let us now return to our original example of a parity-violating
decay:
the $\beta$-decay of cobalt atoms, Co$^{60}\rightarrow $Ni$%
^{60}+e^{-}+\bar{\nu}_{e}$. To model this in the electro-weak
theory outlined above, quarks need to be included. In addition to
the doublet under local $SU(2)$ transformations, comprising the
left-handed electron and electron-neutrino fields, there are weak
isospin doublets containing the muon and muon-neutrino, the tau
and tau-neutrino and three left-handed quark doublets. The first
of these comprises the up quark field $u_{L}(x)$ and a linear
combination of the down and strange quark fields $d_{\theta
L}(x)=\left[ \cos \theta _{c}d_{L}(x)+\sin \theta
_{c}s_{L}(x)\right] $. $\theta _{c}$ is the so-called Cabibbo
angle and its significance is not important for our discussion.
The $\beta$-decay in question involves the decay of a neutron
within the Cobalt nucleus into a proton with the emission of an
electron and its antineutrino. This in turn is to be understood as
the decay of a down quark bound within the neutron into an up
quark ($n\sim udd\rightarrow p\sim uud$). The interaction term
responsible for this decay is $J^{\mu \dagger }W_{\mu }^{\dagger
}$ where $J^{\mu \dagger}$ contains the term
$2\overline{e_{L}}\gamma^{\alpha }\nu _{eL}=\bar{e}\gamma^{\alpha
}(1-\gamma _{5})\nu _{e}$ as before and also $2\cos \theta
_{c}\overline{d_{L}}\gamma ^{\alpha }u_{L}=\cos \theta
_{c}\bar{d}\gamma ^{\alpha }(1-\gamma _{5})u$. The down quark
decays into an up quark emitting a $W^{-}$ boson which decays into
an electron and its antineutrino. But since only left-handed
quarks can couple to the $W$ boson, only left-handed quarks are
involved in such decays and only left-handed electrons and
positive helicity, right-handed antineutrinos are observed as a
result.

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\end{document}
