Published in Philosophy of Science 74: 452-480 (2007).

Mirroring as an A Priori Symmetry
Simon Saunders1

Abstract. A relationist will account for the use of �left�and �right�
in terms of relative orientations, and other properties and relations
invariant under mirroring. This analysis will apply whenever mir-
roring is a symmetry, so it certainly applies to classical mechanics;
we argue it applies to any physical theory formulated on a manifold:
it is in this sense an a priori symmetry. It should apply in particular
to parity-violating theories in quantum mechanics; mirror symmetry
is only broken in such theories as a special symmetry.

1 Introduction

When I say a symmetry of a theory de�ned on a manifold is a priori, general,
or trivial, I mean it in the sense that translations are symmetries of general
relativity (GTR) even when spacetime, as de�ned by the metrical �eld, lacks
translational symmetry. Thus let g represent the metric tensorand T the energy-
momentum tensor for the non-gravitational �elds, de�ned on a manifold M.
For simplicity, let M be di¤eomorphic to R4, so that a single set of rectilinear
coordinates xk : M ! R, k = 0;1;2;3 su¢ ces. Let t : M ! M represent
a translation, i.e. for each q 2 M, it induces the action xk(q) ! xk(t(q)) =
xk(q)+const on the coordinates xk. Then, since the �eld equations of GTR are
generally covariant, if hM;g;Ti is a model of GTR so is hM; t�g;t�Ti, where
t�T; t�g are the pull-back of T and g under t (where, in the simplest case of a
scalar �eld, t��(q) = �(t(q)).)
There are of course in�nitely many such symmetries of theories on manifolds;

the same is true of any smooth transformation on M. It is trivial translational
symmetry, in contrast to the substantive, non-trivial kind, when translations
are isometries of the metric, i.e. when they satisfy t�g = g. For t an isometry,
if hM;g;Ti is a spacetime model in GTR, so is hM;g;t�Ti. This condition is
stringent: it says the metric (and, in GTR, the matter distribution) is perfectly
homogeneous. Such symmetries are usually called �special�.
Whether special or of the trivial, a priori kind, symmetries threaten a form

of underdetermination�they yield distinct representations, hence seeming states
of a¤airs, where no observational distinctions are to be found. �Relationism�
as I shall understand it is a general framework for avoiding such underdeter-
mination: it says that only quantities invariant under exact symmetries are

1To contact the author, please write to: Philosophy Centre, Oxford University, 10 Merton
St., Oxford, Oxen OX1 4JJ, UK; e-mail: simon.saunders@philosophy.ox.ac.uk

1



real�thus relative directions, relative distances, and so on, under rotations and
translations etc. A principle like this has been implicit or explicit in most re-
lationist and more generally �structuralist�interpretations of physical theories,
and not only spacetime theories (and not only, for that matter, physical theo-
ries, see e.g. Nozick 2001). Call it the invariance principle. The distinctions
among representations then correspond to nothing physically real. Equivalently,
such models, as goes their physical content, can be simply identi�ed. They are
Leibniz equivalent (LE), in the terminology of Earman and Norton (1987).
Every symmetry transformation transforms something relative to something

else. In the case of special symmetries certain physical �elds T are dragged
relative to others g, and the existence of the symmetry re�ects a symmetry of
T or g�something that might well be only contingently symmetric. By contrast
trivial symmetries signal the symmetry of something much more abstract: sym-
metries of the bare manifold M, equipped only with its atlas of charts. As such
it applies to any theory formulated on a manifold, just as do the di¤eomorphism
symmetries inherent in the concept of manifold.
I take all of this to be tolerably clear. But its extension, or lack of it, to

the discrete symmetries�to permutations and inversions (spatial, temporal, and
of matter and antimatter)�is obscure. Here we shall concentrate entirely on
spatial inversions in odd dimensions (or mirroring in even ones�we shall use
the terms interchangeably). Our claim is that neither mirror-symmetry, nor
its violation in quantum mechanics (parity violation), o¤ers any impediment
to this program. Invariant quantities (whether under general or special mirror
symmetry) are su¢ cient to account for the appearances and even the workings
of theories, here as in the case of continuous symmetries.
To keep things as simple as possible, we shall restrict attention throughout

to orientable manifolds. In that case an atlas of charts on M can be chosen so
that every chart is similarly handed (so where charts x, x0 overlap, the Jacobean
of the matrix of derivatives @x0j=@xk is positive). There are two such (di¤eo-
morphism classes of) atlases, call them the left-atlas and the right-atlas. Let r
be a manifold homeomorphism inducing the transformation from the left-atlas
to the right-atlas and vice versa (see §4 for a more precise de�nition). Our claim
is that if hM;g;Ti is a model of GTR so is hr(M);g;Ti, and equivalently, so is
hM;r�g;r�Ti; and that the same is true of any other physical theory formulated
on a manifold. By the invariance principle (or LE) the two should be identi�ed
as representing the same physical state of a¤airs.
From a formal point of view that accounts for general mirror symmetry.

However special mirror symmetry, and speci�cally the breaking of this symmetry
in particle physics, poses a harder problem: it can hardly be that T cannot be
re�ected relative to g, taking g as the Minkowski space metric, as the latter has
special mirror symmetry, so that by trivial mirror symmetry it can. Most of
what follows is directed to this question.
To this end we make use of the core relationist treatment of handedness

and parity violation to be found in the writings of Earman (1989), Hoefer
(2000), Huggett (1999, 2000), and Pooley (2003)), but with departures from
their broader perspective, which sees relationism as committed not just to the

2



invariance principle of LE but to the non-existence of space, a speci�cally meta-
physical doctrine that may require changes in physics (forcing, say, a Machian
theory of gravity, possibly at variance with GTR). Relationism, or structural-
ism, as we understand it, as imposed by the invariance principle, is in no sense so
committed. It is rather a framework for the interpretation of physical theories�it
is �non-reductive relationism�, in the terminology of Saunders (2003).
If it is indi¤erent to one popular context of the debate over relationism, it is

sensitive to another: the provision of a uniform approach to the failure of special
symmetries for theories on manifolds. The standard device is the relative drag,
as already illustrated in the case of translations, where one �eld is dragged
relative to another, yielding a model which is unphysical. But in the face of
parity violation in Minkowski space, this cannot be expressed by the relative
drag of matter and space, as we have seen. If we are sticking to the manifold
formalism, and do not move directly to Hilbert-space structures, de�ning parity
violation in terms of the non-commutativity of the parity operator with the
Hamiltonian�a strategy we shall in due course consider�presumably we should
�nd particular handed, material structures, that cannot be inverted relative to
each other, as a matter of law.
What are those structures? How might they be used in an epistemic con-

text? (that is, in the context in which the ordinary human hand is used)? But
questions like this run against the tenor of conventional relationism, which is in-
tent on denying that chiral standards have any signi�cant intrinsic nature, over
and above the fact that they are handed. Thus Pooley (2003, 259-60) holds
that the only thing that distinguishes a standard for �left�from other handed
objects is the fact that it is called �left�. He gives the example of proper names:
no more is there anything about Immanuel Kant that made it correct to call
him �Immanuel Kant�other than that he was called �Immanuel Kant�. That is
I think the right analysis of how ordinary talk about handedness gets o¤ the
ground, but we are interested in something else: what might count as a stan-
dard for �left�that can be used with modal force. A concern of this kind has
been expressed before (by Huggett [2000, 235] and Hoefer [2000, 253]), but to a
di¤erent purpose�the question of how the parity violating laws are to be de�ned
as having one sense, rather than the other. This question, we shall eventually
conclude, is entirely empty.
The idea of a standard that can be used in modal contexts deserves some

new terminology. A standard is internal if it is modelled explicitly, and external
otherwise. Evidently the mere fact that a system is modelled in relation to
others does not give those relations a modal force of the sort we are looking
for (these relations may be entirely contingent), and to mark this point we
shall sometimes speak of a dynamical internal standard; but in principle to be
modelled at all is to be subject to physical laws. We are not interested in �purely
kinematic�models (if there are such; see Brown [2005]).
The distinction between �internal�and �external�standard (and, by exten-

sion, determination) is to be sharply distinguished from that between an �in-
trinsic�and �extrinsic�standard or determination, which is widely in use in the
literature on handedness. It is much closer to Stachel�s (1993) distinction be-

3



tween �nondynamical�and �dynamical�determinations, or �individuating �elds�
(in spacetime theories), and to Bartlett et al.�s (2005) distinction between �non-
implicated�and �implicated�coordinates (in quantum mechanics). See Saunders
(2003) for further discussion of Stachel�s distinction.
From our point of view the use of an external standard is essentially a cop-

out. An external (nondynamical, non-implicated) standard has to come from
somewhere, a supposed background physical system not itself modeled explicitly.
Such a background system is always available, as long as the model is restricted
to a subsystem of the universe, and the symmetry, or lack of it, is then a sub-
system symmetry (in the terminology of Brown and Sypel [1986]). But that is
problematic in cosmology; �relationism�in our sense may not be reductionism
(about space) but it is certainly a response to the underdetermination questions
that preoccupied Leibniz in his correspondence with Clarke. These questions
were always cosmological in scope. Correspondingly, the symmetries of relevance
to us are always global transformations of closed physical systems (and the
invariance principle and LE apply only to these).
Our program, in summary, is this: when mirroring is not a special symmetry,

to give an internal, dynamical model of handed systems, the relative re�ection
of which is dynamically excluded. First, however, we consider the situation in
classical mechanics, where mirroring is a special symmetry.

2 Mirror Symmetry in Classical Mechanics

It seems obvious that if a con�guration of particles is handed, it must be handed
in one way or the other, which way being reversed on mirroring. It is further
clear that mirroring is a symmetry of classical mechanical theories. In such
theories, then:

1. Mirroring is a symmetry.

2. A handed con�guration of particles or �elds is either left-handed or right-
handed and is reversed by mirroring.

By the invariance principle it follows from (1) and (2) that e.g. �being left-
handed�is not a real physical property, an apparent absurdity. In terms of LE,
it follows from (1) and (2) that the world and its mirror image are the same, and
hence that left-handed and right-handed things are the same, again an apparent
absurdity.
This puzzle dates back to Kant�s argument from incongruent counterparts

thatwas in turn inspiredbytheargumentsof theLeibniz-Clarkecorrespondence.
The great di¤erence is that whereas in the case of translations or rotations a
given state of a¤airs and its translate or rotate will at least look just the same,
when considered in themselves, that is no longer true when it comes to mirroring.

4



2.1 Global and subsystem mirroring

(2) can hardly be faulted when mirroring is a subsystem symmetry, when
only a subsystem is subject to a symmetry transformation (under an �active�
interpretation�as when Galileo�s ship is put out to sea), but it is in question
when everything is mirrored. For consider:

(i) Let everything and everyone, including language users, be mirrored. Since
it is a symmetry, the result will be physically possible if the original was.
But the linguistic, neurological behaviour in this mirror-image world will
be functionally the same. Di¤erences, if there are any, will be inexpress-
ible.

My doppelganger�s speech patterns, as longitudinal vibrations in the air, are the
mirror-images of mine, but that does not make for a di¤erence in what he says.
Nor, if inexpressible, should we admit there is a di¤erence in the two cases in
what he perceives or in what he believes. It is independently implausible that
because his neural processes are the mirror images of mine (and are otherwise
identical) that he will perceive things di¤erently from me or understand things
di¤erently from me. The two worlds will be indiscernible from within. (An
argument of this sort was �rst made by Earman [1989, 145-46].)
The same doubt can be raised independent of thought or language:

(ii) The Cinderella experiment: given a handed object, and an experiment
that determines it to be �left�, the mirror-image experiment will determine
the mirror-image object to be �left�as well.

The apparatus incorporates a handed object that can be compared with the
object to be tested (say a shoe, or a glove, or a screw). Thus if the shoe �ts, a
bell is rung. A foot is presented and the bell is rung. Now consider the mirror-
image of this apparatus, presented with the mirror-image foot. Still the bell is
rung! Orientation as measured by a Cinderella experiment is invariant under
global mirroring.
These arguments raise doubts as to whether �left�and �right�can really di¤er

intrinsically at all. For further well-known arguments to this end, that bring
in the dependence of the orientation of a body on the dimensionality or ori-
entability of the space in which it is embedded, see Pooley (2003 p.252-62).
On this basis we can simply reject (2). But there is a prior question here: of
course there are standards for the correct use of �left�and �right�, which advert
to concrete handed systems. What systems, exactly? What is the Cinderella
apparatus, to put the matter in operational terms? It will include a handed
structure for comparison with others (by relations invariant under mirroring),
but it must include something else; the buck has to stop somewhere.
This �something else�cannot of course reside in being handed in one particu-

lar way rather than the other, since �being handed in one particular way�is not
invariant under mirroring, and hence, by the invariance principle, is not a phys-
ically real property. We conclude that it has some distinguishing, invariantly

5



described, and therefore nongeometric property (or if it is a geometric property,
it is one that is invariant under mirroring).
Before considering in more detail what it is, it is worth saying that the

relationist of my stripe need not be committed to denying that �left�and �right�
are genuinely meaningful as purely mathematical concepts. Platonists may well
maintain that a right-handed Cartesian coordinate system is a perfectly good
abstract singular object. And unlike the case of the continuous symmetries, it is
rather more plausible to suppose one has a direct intuition in this case: a state
of rest, in Platonic heaven, is hard to take seriously, and impossible to make use
of, but left-handed coordinates seem to be there for the taking and available for
inspection in the mind�s eye.
We should grant the possibility rather than squash it outright. By all means,

suppose a left-handed atlas of charts exists in an abstract sense; suppose even
that it can be an object of singular thought. The isomorphism claim�that any
mathematical structure built up from the left-hand atlas can be built up from
the right-hand atlas as well�is an entirely independent matter. In no sense
need the invariance principle (or LE) apply to abstract mathematical objects.
But the one condition we do insist on is this: any such perception of abstract
mathematical objects is to be sharply distinguished from any visual, sensible
perception of physical objects. The point is not that there must be an indivisible
gap between the abstract and concrete (that we can leave as an open question),
but that we have already taken care of sensible perception of physical objects:
they, and if necessary the observer, are to be modeled explicitly (the object as
a standard is an internal, embedded standard).

2.2 Left and right as internal standards

To summarize, the relationist needs only to reconcile

MS1 Mirroring is a global symmetry

which rules out the use of an external standard, with:

MS2 A handed object is determinately (measurably) either left-handed or right-
handed, and is invariant under global mirroring.

We may break (MS2) into two parts. First, purely geometric determinations
of relations of congruence and incongruence among handed objects (for conve-
nience, call �enantiomers�). This is unproblematic in the case of an orientable
space (and the ambiguities introduced in a non-orientable space are easily re-
moved):

MS2a An enatiomer is left-handed or right-handed as determined by its congru-
ence and incongruence relations with other enantiomers.

Such determinations are clearly invariant under mirroring.
However the buck must stop somewhere. (MS2) also involves:

6



MS2b Certainenantiomersaredeterminedas left-handed, andothers right-handed,
by virtue of some nongeometrical feature of such objects.

With that, so long as the �certain standards�mentioned in (MS2b) are incor-
porated into the physical description, one has a characterization of �left�and
�right�as internal determinations.
According to Pooley, all that is at issue with (MS2b) is use of language.

The additional non-geometric feature of such objects is just that they reliably
get called �left�or �right�, on what basis no matter. And if one looks at how
the use of �left�and �right�is actually taught, it would seem that any system
of rules would do, linking certain sorts of handed objects with some standard;
all that would be needed of the standard over and above the fact that it is
handed is that (as a purely practical matter) it had no incongruent counterpart,
or if it does have an incongruent counterpart, that it be recognizably distinct
from it (say, that it be di¤erently colored). In practise, the human hand on
the side of the heart does very well (since most human hearts are on the same
side of the body); but so would the side of a car near the steering wheel, at
least in certain countries; the side of the pedals furthest from the accelerator,
in all countries. It would be absurd to single out any one of these features (all
invariantly described) as the feature that makes its bearer left-handed. It could
even be a certain muscular twitch that I use to tell me which of my hands is
right�meaning, which is the one I have been taught to call right, in conformity
with others�use of the term.
With all of this we should agree, as goes our ordinary use of the terms �left�,

�right�. It is certainly consistent with the invariance principle: whatever these
criteria, they are invariant under global mirror imaging. Nor, up to this point,
is the situation really so unfamiliar. Consider the case of length. Time was
when the de�nition of length in SI units was given by congruence relations with
a particular rigid object, maintained under controlled conditions in a de�nite
place (a platinum bar, located in a vault in Paris). We may imagine, in the same
vault, a particular glove as well (say, a boxing glove, that cannot be turned inside
out). To be one metre long just is to have the same length as that particular
bar (picked out by features independent of its length). Is our proposal not just
the same? Even the obvious di¤erence with length, that there is in fact no need
for any physical glove, is easily explained: chiral relations are 2-valued, unlike
relative distances, so there is no need for precision; they can be told at a glance;
is there any more to the di¤erence between chiral and distance relations than
this?
Suppose, indeed, that only the orders of magnitude of relative lengths were

ever required: then there would be no need of the platinum rule either. And
in practise that is what we still do, making do with the foot, the arm�s length,
and the �nger�s breadth, relative dimensions that can be judged at a glance.
Telling the approximate height of a man in feet is no di¤erent from telling his
left hand from his right. It is the same with all other familiar determinations, for
weight, temperature, colour and smell: they all begin as qualitative judgments.
The di¤erence, on this line of thought, is only between judgments that allow

7



for complex logical comparisons (geometry and number), from those that don�t
(colours and smells).
All this �ts with the suggestion, in the case of mirror-symmetry, that the

standard is ultimately singled out on the basis of some nongeometric quantity;
and with the suggestion, further, that this nongeometrical property can be en-
tirely arbitrary. Thus the particular bar of platinum that was used as a standard
of length was essentially chosen arbitrarily. When it comes to the choice of a
standard glove to put alongside it, that too could be wholly arbitrary. But it is
equally clear that these standards, when modeled as internal to the total system,
do not exactly lend themselves to this task. Platinum bars and boxing gloves
are hardly natural systems to include in a dynamical model of an atom, or a
�uid, or a crystal. Just for that reason, no macroscopic bar could ever provide
a very precise standard that could actually be used�and the platinum bar did
not long stay the standard.
A �rst step in improving the accuracy of length determinations was to control

for �uctuations in its length (of how it was mounted and at what temperature
and pressure, controlled in what way). This was to some extent a matter of
modelling the bar dynamically, but proved a complicated matter, insu¢ ciently
accurate, not only in controlling for variations in the bar itself but in using it
to calibrate other rules. In 1960 the SI unit of length was changed to a cer-
tain number of wavelengths of an emission line of Krypton 86, a standard that
could be used to calibrate other emission lines very easily.2 There remained
non-geometrical features to it of course�the particular isotope and the partic-
ular atomic transition process�as well as purely numerical conventions (the
number of wavelengths chosen), but �x on the latter and the standard has a
clear dynamical characterization (the minimal theory to do this in this case is
quantum electrodynamics [QED]).
Is it a stretch to similarly seek an internal, dynamical standard in the case

of the discrete units�say a sample of positive (rather than negative) charge, as
we have a sample of left-handed (rather than right-handed) hands? Surely not,
if we include time-inversion along with mirroring and charge-conjugation, the
family to which it belongs. The goal of �nding an internal, dynamical standard
of the forward (rather than backward) arrow of time is widely shared.
The obstacle to de�ning internal standards in simple dynamical terms, in

the case of the discrete symmetries, is not that they are not needed but that
they seem impossible to �nd. There seems to be no simple classical mechanical
model of �the positive �(charge) or �the left�(handedness) or for that matter of
�the forward direction in time�. Worse, there may be no complicated classical
model either, no internal dynamical standard of any description. For how can
any of those things (the positive, the left-hand, the future arrow) be dynamically
characterized, if on inversion the dynamics is the same? Just here there is an
important contrast with the case of length: there is indeed a dynamical, internal
standard of length, in QED, that can couple to other physical systems, but QED

2It was changed again in 1983, when it was subordinated to a de�nition of the second (and
de�ned in tems of the distance travelled by light in a certain fraction of a second).

8



does not have scaling as a special symmetry.
The implication, restricted to mirror-symmetry, seems to be clear. The

relationist strategy, in the face of underdetermination with respect to mirroring,
is to identify a model and its mirror-image just as it identi�es models related by
continuous symmetries. Invariant quantities alone are shown to be su¢ cient to
characterize a certain class of things as left-handed, on the basis of their intrinsic
properties. The method works because of a trick�these internal properties are
independent of geometry per se. But the price is that they appear wholly
arbitrary; they are whatever language use happened to most conveniently �x
onto. Just insofar as mirroring is a symmetry, it seems the internal standard
cannot be characterized dynamically.

3 Parity violation

The situation is transformed in moving to a parity-violating theory, like the
electroweak theory in quantum mechanics. The way is then open to de�ning
a chiral standard that is not just internal, but non-arbitrary too�a genuinely
dynamical marker. But then, if mirroring isn�t a symmetry, why isn�t the en-
tire relationist program simply an irrelevancy? For if mirroring is no longer a
symmetry there is no longer a threat of underdetermination.
The answer is that mirror symmetry remains as an a priori symmetry, and

poses just as great a threat of underdetermination in that guise. Nothing in
the previous section (except at the very end) hinged on the di¤erence between
special and a priori or trivial mirror symmetry.
To begin, we should learn how parity violation is experimentally manifested,

and how it is represented in the relevant equations (quantum electroweak theory
or QET). This ground has been covered by Earman (1989) and in more detail
by Pooley (2003), but their basic points need repeating and, in one instance
(the de�nition of a spin structure on a manifold) they need to be extended.

3.1 Evidence for parity violation

We follow Earman in considering the experiment of Crawford et al. (1957).
It was observed that when a negative pi meson, ��, scatters o¤ a proton, at
certain energies it decays into two neutral particles (the hyperon, �0, and the
neutral K�meson, K0), de�ning a plane. The hyperon subsequently decays
into another pi meson �� and a proton, which lie on (in general) a di¤erent
plane. Now take the mirror image of this process through the �rst of the two
planes, so whereas before, e.g., if the �nal �� particle moved out of the plane
going down, in the mirror-image process, it moves out of the plane going up. It
turns out that processes of the second sort are much rarer than processes of the
�rst sort (see Figure1).

Figure 1.�Parity violation for pi meson decay.

9



10



We may quantify the result in terms of the cross product ��!p��
in
� �!p� of the

momentum vectors of the hyperon and the pion present before the second decay.
Let the momentum vector of the �nal pion be ��!p��

fi
. Then the two processes,

the one the mirror-image of the other, can each be de�ned as a condition on the
pseudoscalars �

��!p��
in
��!p�

�
���!p��

fi
> 0 (1a)

�
��!p��

in
��!p�

�
���!p��

fi
< 0 (1b)

Since this is quantum mechanics, the e¤ect is only statistical, but nevertheless
real: one of them (the positive value) is preferentially observed over the other
(so [1a] is the correct equation for this kind of decay). Correspondingly, it is
not preserved under mirroring (which interchanges [1a] and [1b]).
The cross-product in turn is normally de�ned, in elementary texts, with

the aid of a hand-rule, either explicitly using one or other of the small family
of chiral terms, �clockwise�, �screw�, �right-hand�, or by means of a diagram.
Quantitatively, if � is the angle between two vectors

�!
A;
�!
B , of magnitude A and

B respectively,
�!
A �

�!
B is a vector of magnitude AB sin�, orthogonal to the

plane de�ned by
�!
A and

�!
B . It�s direction is �xed by Figure 2. This picture is

not itself modelled in the equations �it is an external standard.

Figure 2.�The right-hand rule.

Insofar as the phenomenological equations go, and in agreement with the
account of §2, �left�and �right�are being used as external determinations. This
is consistent with taking mirror symmetry as a subsystem symmetry, and its
violation as the violation of a subsystem symmetry.

3.2 Electroweak theory

In a certain sense, in quantum mechanics, any application of the theory, any
symmetry, can only be a subsystem symmetry, for the theory requires an outside
observer or experimental context that is not itself a part of the system that is

11



modelled. Quantum states  are only attributed to subsystems, and their uni-
tary evolutions under Hamiltonians H are only applied to subsystems. There is
no �wave-function to the universe�: the observer, in quantum mechanics, cannot
be �naturalized�.
All of these statements are of course highly contentious�the �certain sense�

was Bohr�s view of the matter. The three leading realist alternatives, hidden-
variable theories, dynamical collapse theories, and the Everett interpretation
(where only the latter leaves the equations of quantum mechanics unchanged),
do suppose there is a wave-function for the universe and are perfectly able to
naturalize �the observer�. But for the moment we are asking about the equations
of QET as ordinarily applied to parity violating systems, which are undoubtedly
modelled as sub-systems of the universe. We are talking of the violation of a
sub-system symmetry.
As ordinarily applied, the condition for mirroring to be a symmetry in quan-

tum mechanics is that the Hamiltonian, H, commute with the inversion opera-
tor, P (a unitary operator); it is broken as a symmetry if [P;H] 6= 0: This means
that if  (t) is a solution to the Schrödinger equation with the Hamiltonian H,
then P (t), is not such a solution, and does not represent a physically possible
evolution of the subsystem.
It seems we have everything we desire, save for the fact that these states of

a¤airs (or disallowed states of a¤airs) are de�ned relative to a given background,
the classical world, classically described experiments, which are all external
standards. The analysis is immediately problematic as soon as we insist, as we
must, that quantum mechanics be applied to a strictly closed system (with the
observer modelled as within). This, the direct route, is hamstrung by questions
in the foundations of quantum mechanics, not least the problem of measurement.
But it appears that for present purposes we can sidestep these questions.

We can go a considerable way to understanding the nature of parity violation
by considering the structure of the parity-violating Lagrangian as a system of
�elds (quantum or classical) de�ned on a manifold, to the neglect of the state.
(In �eld theory one anyway works in the Heisenberg picture, in which the state,
independent of measurement, does not change.)
In this spirit, consider the idealized case in which all the fermions are treated

as massless. The parity violating part of the interaction Lagrangian is3:

EL(iD)EL = ELi
�(@� � igAa��

a �
i

6
g0B�)EL: (2)

This is a Lorentz invariant, U(1) and SU(2) gauge invariant, self-adjoint scalar.
D is the covariant derivative; it is a rule to de�ne a derivative, by which one
can compare the phase of the various �elds at di¤erent points in space. Since
the rule is path-dependent, or non-integrable, it introduces gauge �elds into
the theory (the �elds Aa�; B�). Here B� for � = 0;1;2;3 are the components
of a U(1) gauge �eld (that enter into the photon coupling as well as that of

3My notation follows Peskin and Schroeder (1995 equation 20.78), except that I have
omitted the right-handed chiral �elds (which are present in QED and QCD, but not in EQT,
with the same couplings as their left-handed partners to the gauge �elds).

12



the neutral vector boson W0) and Aa� those of an SU(2) gauge �eld symmetry
(one vector �eld for each of its generators, and two of the three associated with
the charged vector bosons, the W� particles). EL is the left-handed fermion
doublet (�e ,e� )L, where �e and e� are the neutrino and electron spinor �elds.
Because massless, the fermion doublet has a unique frame-independent helicity,
which we use to label it as �L�or �R�respectively. Only if a theory contains
both kinds of fermions (or fermion doublets) can it host a representation of the
full Poincaré (or �improper�Lorentz) group, which includes the inversions.
The absence of right-handed fermions from this theory shows rather vividly

that it violates mirror symmetry, as the action of the inversion operator P is
to interchange them. In fact, it is su¢ cient (and this is how it works in more
realistic models) for the two chiral spinor doublets to couple to gauge �elds of
di¤erent symmetries, or to inequivalent representations of the same symmetry.
The kinetic energy term for Dirac fermions  can always be split into separate
pieces for the left- and right-handed �elds:

 i
�@� =  Li
�@� L +  Ri

�@� R:

The derivatives can then be replaced by covariant derivatives using di¤erent
gauge groups or di¤erent representations of the same gauge group. The result
will be a P�violating theory.
The key question concerns the use of �left�and �right�as designations of the

spinor �elds. They have a clear geometric meaning in terms of helicities�the
projection of the component of spin in the direction of motion�but a meaning
�xed by the use of a hand rule, either the left-hand rule or the right. Do they
have an independent algebraic signi�cance? As Dirac bi-spinors they are de�ned
by the projections

P� = 1� 
5 = 1� 
0
1
2
3 (3)

where P+ =  L, P� =  R. The Dirac 
�matrices are 4 � 4 matrices that
transform as a Lorentz vector and obey the anticommutation relations:


�
� + 
�
� = 2g��: (4)

The change from a left-handed to a right-handed coordinate system interchanges
two of the 
�matrices (for �,� 2 f1;2;3g), and therefore interchanges P� and
P�. The same can be achieved by a change of representation of these matrices,
consistent with their transformation properties under the Lorentz group and the
algebra of (4).
The same point can be made in terms of two-component spinors. Quite

generally, when introducing spinors on a manifold, one needs a spin structure�a
correspondence between tensors acting in the tangent space Tq at each point
q 2 M and endormorphisms acting on a complex vector space Vq at q�linear
transformations of spinors at q. In the case of the Lorentz group, the funda-
mental connection is between a second order Hermitian spinor �eld � :

AB
(dotted

indices are complex conjugates of undotted ones), acting on a 4�dimensional

13



complex space and a real vector �eld, A, on Minkowski space, as given by four
real functions A�, � = 0;1;2;3, thus:

� :
AB

= �
�
:
AB
A�: (5)

Here the four matrices ��:
AB

are Hermitian and transform as a second-order

spinor in the indices
:

A;B and as a Lorentz four-vector in the index � (my
notation is standard, following e.g. Davis [1970, 281-83]). These matrices are,
of course, the Pauli spin matrices (with the � = 0 matrix given by the identity),
where their spin indices refer to their transformation properties under SL(2;C),
the unimodular group of complex linear transformations in 2 dimensions, the
covering group of the proper Lorentz group.
The Pauli matrices, just like the 
 matrices, anticommute for � 6= �; mirror-

ing interchanges two of them with a corresponding change in sign. The choice
of one sign rather than the other indicates one of the two spinor representa-
tions (that are interchanged under mirroring) one is using. Thus, for the free
zero-mass neutrino �eld, one has the two possibilities:

(�0:
AB
@0 � �k:

AB
@k)�

B
L = 0 (6a)

(�0:
AB
@0 + �

k
:
AB
@k)�

B
R = 0 (6b)

(together with the complex conjugate equations describing their antiparticles),
corresponding to massless left-handed and right-handed neutrinos respectively.
It is only when there are pairs of spinor �elds of the same mass that one can
de�ne a representation of the full Lorentz group (as with Dirac bispinors).
Which of (6a), (6b) is the right one? The one that gives the right sign for

the measured pseudoscalars �the one that gives (1a) rather than (1b), where
the correctness of the latter is determined by a hand-rule, Figure 2. That is
essentially to de�ne the spin-structure itself in terms of this hand-rule, just as we
did the cross-product. Once this has been done, one can predict the orientation
of every other parity-violating decay process in the scope of the theory. Had
they been equipped with it in advance, Crawford and his collaborators would
have been able to predict which laboratory wall the decay products were headed
for.
But just because the choice of sign is made in this way, it is clear that �left�

and �right�are being used as external determinations. The external standard
(human hands and whatnot) is not itself described by the equations. Rather,
the external standard is used to pick out one of two sets of equations, that are
mapped into one another under mirroring. The situation in these regards is no
di¤erent from the phenomenological equations (1a, 1b).

3.3 The invisibility of mirroring as an a priori symmetry

Evidently we need to go beyond mirror symmetry breaking as de�ned by an
external standard, and mirror symmetry as a subsystem symmetry. How, ex-
actly? Obviously, by modeling these external standards explicitly. But once

14



product

3:pdf

this is done we expect, from the Cinderella experiment, that global mirroring
(including mirroring of the standard) will become invisible. Pseudoscalars will
no longer be measurable.
To rehearse the steps of the argument, but applied to a parity-violating ex-

periment, consider the phenomenological equations (1a), (1b). They are given
a de�nite sense by an external standard, Figure 2. If the hand depicted were
included in the phenomenological description of the decay product, the descrip-
tion would be correct insofar as A is the pion direction, B the meson, and C
the kaon, and the �ngers and thumb of the hand can be correctly aligned with
the decay products, as in Figure 3(a). But now if everything is mirrored that
alignment will remain correct, as shown in Figure 3(b). There can be no pos-
sible evidence internal to worlds that could tell us which of the two worlds is
ours. Correspondingly, the phenomenological equation (1a) becomes invariant
under mirroring.

(b) (a)
Figure 3.�Invisibility of global mirroring as a trivial symmetry.

15



What has just been demonstrated is the invisibility of global mirroring as a
trivial symmetry, where everything is mirrored relative to the atlas of charts,
including what was hitherto used as an external standard.
All well and good, but if we are after a statement of the violation of special

mirror symmetry, we should seek a standard whose relative drag, leaving �xed
the QET matter and gauge �elds of §3.2, yields a model that is not physically
possible: for this purpose human hands will hardly do. Indeed, from a relationist
point of view, the contingency of any particular hand rule is of the sort: the hand
on the side of the typical human heart is congruent to certain decay processes.
Not only is that a bad candidate for a dynamical internal standard, it is not

even clear that the notion of its relative drag is even well-posed. For no hand,
or anything else made up out of ordinary matter, could be spatially inverted
relative to a certain set of quantum �elds, when those very �elds are what
ordinary matter is made of. Such a relative drag would not even make sense as
a mathematical model. it would be ruled out, not because it fails to satisfy the
equations of the theory, but because it is a contradiction in terms.

4 Breaking mirroring as a special global sym-
metry

For simplicity, consider as in the beginning manifolds di¤eomorphic to R4, so
only one chart is needed. Let g be the metric on M and let �(�); � = 0;1;2;3
be a tetrad �eld on M, a quadruple of vector �elds �(�) : M ! T(R4),
g(�(0);�(0)) = 1, g(�(0);�(j)) = 0; g(�(j);�(k)) = ��jk; j;k = 1;2;3. Let
r : M ! M satisfy

��(q) = (2�0� �1)��(r(q)); q 2 M:

This manifold map induces global mirroring of the tetrad �eld. Evidently the
latter can be used to de�ne coordinates x� on M (so that �(�) = @=@x�), on
which r likewise induces global mirroring. In this way we de�ne global mirroring
for every tensor �eld on M.
What about spinor �elds? From our previous discussion (§3.2) there is a

separate stipulation connecting the spin space Vq at q 2 M with the tangent
space Tq, as given by (5). Let us indicate this by �, with the opposite choice
given by �: Let F denote tensor �elds and spin �elds taken together, with the
understanding that for spinor�elds r�F is the inducedpoint-wise transformation
only (with no action on �directions�in spin space, which are looked after by �).
Then a priori (trivial) mirror symmetry is the condition:

if hM;g;F;�i is a model hM;r�g;r�F;�i is a model

or equivalently

if hM;g;F;�i , is a model hr(M);g;F;�i is a model

16



meaning, as before, that if the one is a model of a physically possible world
then so is the other. This is the extension of the statement of a priori mirror
symmetry to apply to spin-structures on manifolds.
Before considering the meaning of special mirror symmetry on such mani-

folds, with an eye to its violation, we should �rst consider how mirroring might
be violated even in the absence of spin-structures. The obvious candidate for
the expression of mirroring as a special symmetry is:

if hM;g;Ti is a model hM;g;r�Ti is a model.

We earlier dismissed the failure of this condition as adequate for characterizing
parity violation, as the latter is de�ned in a Minkowski space theory, but it does
characterize a certain kind of mirror symmetry breaking: any theory in which
mirroring is neither an isometry of g, nor of T , will do. A prime example is
GTR.
That might seem a disappointment: GTR breaks special mirror symmetry

with no need for chiral spinors; any asymmetric con�guration of a single scalar
�eld will do. But thinking again, the implication is welcome, for the failure of
any other special symmetry t in GTR is expressed as the absence of the associ-
ated isometry, t�g 6= g; why suppose that mirroring should be any di¤erent? My
claim throughout is that mirroring is to be treated no di¤erently from any other
symmetry. And anyway, unless a special symmetry is required by a theory, that
theory should not be bound to respect it; GTR requires no special, continuous
spacetime symmetry, so why should it require special mirror symmetry?
On the other hand, one speaks of certain models of GTR as satisfying a sym-

metry, rather than GTR itself�and, by extension, of certain models as violating
a special symmetry. Thus Schwarzchild spacetime has spherical symmetry but
not translational symmetry; Robertson-Walker spacetimes violate time trans-
lation symmetry (they are not static) but they are spatially homogeneous and
isotropic. Many models have mirror symmetry in this sense.
Mirroring, we conclude, whether as a trivial or special symmetry, falls into

place in GTR in exactly the same way as do continuous symmetries. The real
di¤erence is between GTR and other dynamical theories; between the inherent
lack of symmetry of the distribution of �elds on the manifold, as compared to the
inherent symmetries of �elds in the tangent space at each point of the manifold.
This is the arena of all the non-gravitational physics, so it is no surprise that
symmetry-breaking at this level has a fundamentally di¤erent character from
that in GTR. It is su¢ cient therefore to consider only Minkowski space theories.

4.1 Minkowski-space theories

Theories on Minkowski space are of course theories on manifolds, so we may
continue with the same notation, with the di¤erence that the metric g is �xed
once and for all as that of Minkowski space, denoted �. It is homogeneous,
isotropic, and mirror symmetric.
The condition for special mirror-symmetry breaking (SMB) is then:

17



There is some model hM;�;T;�i such that hM;�;r�T;�i is not a model
(SMB)

(and equivalently, by trivial mirror symmetry, that hM;g;T;�i is not a model).
In fact (SMB) would appear to be maximally violated in the presence of the elec-
troweak interaction: for no model hM;g;T;�i ; is hM;g;r�T;�i a model. And
evidently what is ruled out is mathematically well-de�ned�there is no contra-
diction in re�ecting (relatively dragging) only spinor �elds at each point q 2 M,
and not tensor �elds as well. Rather, one ends up with relative con�gurations
of �elds in spacetime that do not satisfy the equations of QET.
But how are the two descriptions to be compared�how are the equations for

the whole system to be set up, that make use of no external standard? Will they
not also make use of a cross-product, as with (6a), (6b), or the equivalent choice
of representation of the 
�matrices? Indeed, but we have already granted this
point. By all means make use of a purely abstract standard for �the left�, say
the left atlas of charts. The relationist claim is that any such abstract standard
is ex hypothesis irrelevant to the distinguishing feature, whatever it is, that
determines the standard for �the left�within the model (the internal standard),
the standard whose congruence and anticongruence relations with other handed
bodies determines the use of all other chiral terms. Further, the relationist�s
presupposition is that any body of mathematics that can be built up using the
left atlas can equally be built up using the right atlas�that the two structures
will be a priori isomorphic. That is to say, by trivial mirror symmetry, either
(6a) or (6b) can be used in setting up the total system of equations; it makes
no di¤erence. The total system of equations in either case will be isomorphic.
So far, and for the reasons mentioned at the beginning of §3.2, we have made

no mention of Hilbert space structures or the quantum state. But trivial mirror
symmetry has a reasonably clear expression in these terms: it is that, if P is the
unitary operator representing spatial inversion (in odd dimensions) on a Hilbert
space H, on which the Hamiltonian is the operator H, and the state 	 describes
the total system, then the one system of equations is hH, H;	i, whilst the other
is hH, PHP�1;P	i. That makes the triviality of the claim perfectly obvious.
But that also makes it clear that it is a mistake to suppose that parity violation
amounts to the statement that whilst 	(t) is a possible history of the closed
system, P	(t) is not (for we have just granted that it is).
Returning to the system of �elds, mirroring is a special symmetry if all of

the spinor �elds can be re�ected, but none of the tensor �elds. That would
require that the spinor �elds themselves carry a representation of the group of
spatial inversions. We have already seen that this is not so in QET, although
it is true of QED and quantum chromodynamics. But whilst that expresses the
failure of mirroring as a special symmetry in QET, it does not give us all that we
were promised. The suggestion was that we could �nd an internal, dynamical
standard for our use of chiral terms, with lawlike congruence relations with other
systems and �elds; so far, it seems all that is on o¤er are aspects of the structure
of QET itself.

18



4.2 Dynamical internal standards, again

We are asking for something to �ll the role of (MS2b), something that is nat-
urally invariant under mirroring as an a priori symmetry, something that can
actually be used. What is needed is a standard sensitive to the electroweak
force, to which it couples directly, but which is at the same time observable or
at least indirectly observable. That suggests the standard should be close to
microscopic. Very simple examples at the ultra-microscopic would be di¤erent
kinds of parity-violating decay processes�whose relative orientations could not
be reversed consistent with dynamical laws�but much better would be a stable
quantum system, not so large so as to unmanageably complicated, nor too small
or too �eeting to function even in principle as an observable standard.
The obvious place to look is chiral chemistry. Indeed, very soon after the

fall of parity, lawlike e¤ects of this kind were conjectured. Thus Ulbricht (1959)
argued that where � rays could in�uence the decomposition of enantiomeric
molecules, they would do so preferentially. Irradiation by radionuclides, he spec-
ulated, would produce di¤erential decompositions of enantiomers in a racemic
mixture, with one type more abundant than the other. It was the �rst in a long
line of investigations into the in�uence of the electroweak force on the behaviour
of atoms and molecules.
The e¤ect conjectured by Ulbricht turned out to be negligible, but the con-

tribution to the Hamiltonians of enantiomeric molecules due to the weak neutral
current is not. The resulting di¤erences in binding energies lead to observably
di¤erent thermal reactions for these isomers (Rein 1974, Letokhov 1975). For
the order of magnitude, the change in sign in the contribution to the total en-
ergy of an enantiomer depending on its relative orientation to the electroweak
current is:

�E � �Z5 �10�19eV

where Z is the atomic number of the molecule and � is the fractional dissym-
metry factor (Zel�dovich et al 1977). In the example of Hegstrom et al. (1979),
a triatomic sulphide chromophore C-S-C in the chiral �eld of other atoms of
the steroid gives a fractional factor � � 10�8: The same authors report much
larger factors (of order 10�4 ) for certain ethylenes, where the chromophores are
inherently monochiral.
Some took these results to show that the electroweak force explains the

observed homochiral biochemistry of the Earth (as speculated by Mason and
Tranter [1984]), rather than putting it down to a quirk of evolution. That would
be icing on the cake. For our purposes, it is enough that the relative orientations
that it produces with certain chiral molecules are lawlike. Thus in (MS2b),
the mentioned enantiomers may be taken as, e.g., sulphide chromophores, in
the lowest of the two ground states. The relative orientation of these, with
the decay products of hyperon decay as depicted in Figure 3, is lawlike, and
their relative inversion dynamically prohibited. This will do as an internal,
dynamically embedded, standard of orientation.

19



Still, thiskindof internaldetermination, as specifying the relativeorientation
of something to something else, seems roundabout. Isn�t it possible to short-
circuit this procedure, and just talk of the electroweak currents themselves, as
the enantiomers mentioned in (MS2b)? It may be epistemically further removed,
but the orientations of sulphide chromophores are hardly directly observable.
Aren�t we simply displaying ways in which the weak interaction manifests itself
in molecular physics?
A similar objection can be made to our formulation of (SMB) in GTR: do

not the matter �elds and the metric �eld in this theory mesh so tightly that
obviously they have the same orientation, if oriented at all? Why, to express
mirror-symmetry breaking in GTR, do we have to speak of two �elds here, and
not one?
The answer is that there really do have to be two �elds or structures or

things, and not one, if my two central contentions are true�if mirroring is an
a priori symmetry, and if relationism, in the guise of the invariance principle,
applies to mirroring just as it does to other symmetries. Something handed has
to be compared with something handed, if there is to be a relative orientation,
itself invariant under mirroring, the only meaningful kind. As Weyl said, just
before the discovery of parity violation:

Had God, rather than making �rst a left 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 not in the �rst but in
the second act, by bringing forth a hand which was equally rather
than oppositely oriented to the �rst-created specimen. (Weyl 1952,
21-22).

His claim is just as true today.

4.3 Accounting for contingency

When Pauli quipped in the face of parity violation that �God is weakly left
handed�, the implication, presumably, was that God could have been right-
handed instead. What would the universe have been like in that case?
The easy answer is that whereas (1a) is empirically correct (referred to Figure

2) it could have been (1b) instead. But that is to treat �left�and �right�as
external determinations. The hand depicted in Figure 2, an external standard,
is by �at dynamically decoupled from the electroweak �elds, so of course it
appears only a contingent matter that it has the relative orientation with respect
to decay processes that it has. We have been over this ground already. As an
account of how God could have di¤ered in his designs, it is nothing but hubris. It
is not that God had a distinctive handedness, which could have been di¤erent; it
is that we humans have a distinctive handedness, that could have been di¤erent.
If there is anything to the idea that the fundamental forces could have had

the opposite handedness to the one that they have, we need to consider lawlike
relations between chiral �elds. We should then consider the examples already

20



given�where certain homochiral isomers are dynamically preferred over their
enantiomers, and have such-and-such relative orientation to others. The alter-
native way that parity might have been broken would have led to the opposite
relative orientation of all of those things.
But there is a great deal wrong with this way of putting it. It suggests that

there is an identity of structure in the way things could have been after all �
so we are back to the kind of underdetermination that the invariance principle
and LE were designed to circumvent. And when one looks at the concrete
examples of a relative drag of such a standard, keeping others �xed, or keeping
the geometry of certain electroweak decays �xed, it is an entirely open question
whether there is any theory that could produce such a dynamics�and if there
were one such theory, for a speci�ed range of relative orientations, perhaps there
are many, for that speci�ed range.
Whence the stubborn intuition that there are exactly two possibilities? It

derives, perhaps, from confusing physical distinctions with mathematical ones
(the underlying a priori mirror symmetry). Or, more plausibly, it derives from
confusing global and subsystem symmetries. As a subsystem symmetry, just
because the standard used is not itself dynamically analyzed, it is by de�nition
arbitrary. As a matter of logic there are exactly two possible ways in which the
symmetry could be broken, with nothing to choose between them.
The better way of putting it, if we must speak in Pauli�s terms, is that one of

God�s hands is weaker than the other�and that there are no further facts about
God�s anatomy that distinguish the two. That makes clearer that there are not
two possible worlds he could have created (but in�nitely many, depending on
how much weaker the one hand is than the other).

4.4 Orientation �elds

In recent literature on mirror symmetry and parity violation the suggestion
has often been made that an orientation �eld is needed (Wald 1984, Huggett
(2000)), and that otherwise parity violation is non-locally determined (Hoefer
(2000), Pooley (2003)).
An orientation �eld serves a simple intuition. Imagine a parity-violating

decay takes place in one region of spacetime, and that another, similar, decay
takes place elsewhere, at large spacelike separation. What guarantees that the
two processes are handed in the same way? It had better be something local
to each of the two regions of spacetime, if we are not to have brute distant
correlations; so there must be a locally available standard, varying continuously
from point to point, which directs the dynamics and guarantees (assuming the
manifold is orientable) that the sense of the decay process is everywhere the
same.
But opinions di¤er on what such a �eld could be. Huggett (2000) pursued

an aspect of the absolute-relational controversy that we have been ignoring,
namely, the question of whether inertial or a¢ ne structure is available to a
relationist who denies the reality of space or spacetime, and, if not, what should
be put in its place (essentially, the question of how to implement a Machian

21



dynamical theory). Newtonian principles pick out a certain class of preferred
states, namely, inertial motions: this needed a new structure to underpin it, an
a¢ ne connection. Parity-violating theories likewise pick out a certain class of
preferred states, namely those in which parity is violated with a certain sense:
this too needs some new structure to underpin it. The suggested candidate is
an �orientation �eld�.
In illustration, Huggett (2000) wrote down a simple 1-dimensional 2-particle

quantum mechanical potential function (with � and � positive real numbers):

V (x1;x2) = �(x1 � x2)+ �(x1 � x2)2 = VA + VS

where by construction VA is odd and VS is even under mirroring. If the particles
have the same mass, m, the Schrödinger equation is:

~2

2m
(�r21 �r

2
2 + VA + VS)	 = i

@	

@t
: (7)

It explicitly violates mirror-symmetry. The result is that. e.g., initially sym-
metric (even) states evolve into superpositions of even and odd states. The
equation drives a 2�particle state, initially with a uniform probability density
over a given segment S�S of R2, into one concentrated in a region where x1�x2
is negative. If the negative real line extends to the left, and the positive to the
right, particle 1 will, if found in S, be more probably found to the left of particle
2.
Huggett claims that the distinction between x1 < x2 and x1 > x2, on which

the potential depends, has to be given

In absolute terms, not just relative to some arbitrary coordinates.
Thus, not until an �arrow of space�is given is the theory well-de�ned.
This arrow can tell us for two points whether their separation is pos-
itive or negative�which is the �earlier�spatially speaking�and hence
give de�nite meaning to the Hamiltonian of the theory. (Note the
analogy with Newton�s �rst law: �constant motion�, and thus the
law, is ill de�ned unless some notion of a¢ ne structure is given.)
Of course, once we have observed the development of the particles
we could determine the direction of the arrow, and could express its
direction in relational terms, say by two standard objects and their
order. Once again, the relationist is not faced with a descriptive
problem�or even an epistemological problem�but with formulating
a theory of the process in suitable relational terms, and a plausi-
ble theory should not make fundamental reference to a contingent
standard. (Huggett 2000, p.235).

An orientation �eld, an �arrow to space�, is by contrast more suitable.
There are points of overlap between us. Huggett�s complaint about con-

tingent standards is the same as mine. We agree that the standard should be
internal to the model; but more than that, it should not be any old handed
object; it should not be contingent. He should welcome, then, my suggestion

22



that we choose certain chiral molecules instead. But this sits unhappily with his
comparison of the orientation �eld to an a¢ ne structure: chiral chromophores
are not exactly fundamental in that sense. Nor do they have to be speci�ed in
advance to de�ne the dynamics, on the contrary, they are manifestations of the
dynamics.
The better way of reading Huggett�s remark from our point of view is that

the choice of an �arrow of space�is to be understood as the choice of the left-
atlas, say, rather than the right-atlas. We may grant that only then is the theory
well-de�ned (only then is it written down in a de�nite system of coordinates);
grant too that the choice of the atlas of charts (whether the right-atlas or the
left) should be regarded as endowing the manifold with a particular structure
(call it an orientation �eld rather than a choice of atlas); allow that only thus
is the theory properly formulated in �plausible�and �suitable�relational terms.
But what does not follow is that we have two distinct theories�on the contrary,
our claim is that the choice of an orientation �eld in this sense, whether the left
atlas or the right atlas, makes no di¤erence to the physics.
But Huggett insists there are two distinct theories here, depending on the

choice of the arrow to space, a view backed up by Pooley:

The orientation �eld is either supposed to be a real, physical �eld,
or is supposed to represent some genuinely asymmetric structure
of space or spacetime itself. If this is the case, then one cannot
identify a theory that assigns a certain probability to the vector
from particle 1 to particle 2 being aligned with the arrow de�ned
by the orientation �eld, with a theory that assigns precisely that
probability to the case where the two vectors are in the opposite
alignment. Similarly, a theory that asserts that all electrons which
are �congruent�to an orientation �eld couple to W bosons and those
which are �incongruent�do not, cannot be identi�ed with a theory
that predicts the same phenomena by asserting that all electrons
which are incongruent to an orientation �eld couple to W bosons.
(Pooley 2003, 273-4).

Given Pooley�s presuppositions that the orientation �eld is a real physical �eld
or a genuinely asymmetric structure of space�if the orientation �eld enters into
real (invariant) dynamical relations with other �elds�then assuredly the two
relative alignments of the �elds would be physically distinct. Pooley�s example
of two such distinct alignments�between fermion �elds and bosonic (tensor)
�elds, and oppositely oriented fermion �elds with the same couplings to the
same boson �elds�is also similar to ours. But we are saying that where one is
possible (satis�es the equations of QET) the other is not. The molecule with the
mirror-image spatial geometry of the sulphur chromophore in its lowest energy
ground-state, leaving all else unchanged, is not in its lowest energy ground-state.
A fortiori, if one had two theories which di¤ered in this respect they would not
be the same.
Huggett and Pooley on the contrary see in (7) two distinct theories which

are observationally the same. Why think that? One reason is because one

23



thinks that if mirror symmetry is broken, it must be because one of exactly two
possible worlds (mirror-images of each other) is singled out as real�whereupon
there must be another theory that picks out the other world as real�but that,
as good relationists, the two worlds do not di¤er observationally. This �ts with
the thought, already discussed, that if 	 is a solution to a parity-violating
Hamiltonian H, then P	 presents the mirror-inverted world, and is prohibited
(where U is spatial inversion).
To this our answer is as before: there is only one theory here, with two

isomorphic representations, the one hH;H;	i, the other hH;PHP�1;P	i. Or
it might be better said that this is a parity-violating theory only when referred
to an external standard, where mirroring is broken as a subsystem symmetry.
Indeed it describes only a lone, 2-particle hand, in 1�dimensional space, with
God�s creative decision awaiting the next step, the creation of a second, handed
system.
If there is no natural candidate �orientation �eld�in our framework that gives

Huggett and Pooley all that they want, what of its original rational�to de�ne a
local internal standard? The orientation �eld was to tell, in each local vicinity,
which way particles like hyperons in that vicinity were to decay; otherwise there
would only be �the relationist�s account of parity violation, with its brute, lawlike
non-localities�(Pooley 2003, 274).
The choice of right-atlas (respectively left-atlas) obviously does not serve

this function, for the left-atlas (respectively right-atlas) will reproduce exactly
the same physics. But there is another candidate, surely, which does, namely
the entire system of electroweak �elds. The �elds, we suppose, must be in place
and have the structure that they have �whatever their quantum excitation
numbers�throughout all of spacetime. The �elds do not pop in and out of
existence when particles do.
To be sure, we are back to controversial questions about how, precisely, a

relativistic quantum �eld theory is to be interpreted. In response to this, we
could stick to classical �eld theory. It is true that classical �elds (unlike quantum
�elds) can be strictly zero throughout an open set, and that may suggest they
cannot be relied on to furnish a local standard; but this surely leads to much
more general questions�for example, of whether the local Lagrangian function
is so much as de�ned in a region in which the �elds of which it is a function
disappear.
But even if there is a problem with locality arising in this way, it is far from

clear that it is speci�c to parity-violation. We know that not all goes well in
this respect for gauge theories more generally, classical as well as quantum. The
Aharonov-Bohm e¤ect is a case in point. Here the interference fringe produced
by an electron beam is displaced, depending on the value of the magnetic �ux
through a solenoid in the region of the interfering beams. The values of the
electromagnetic potentials themselves are not real, by the invariance principle,
whereas invariant scalars constructed from the

�!
E and

�!
B �elds all vanish in the

region dynamically accessible to the electrons, the region outside the solenoid.
The e¤ect, from the point of view of invariant local quantities, is holistic, to be

24



described in terms of the holonomy of the �eld�invariant quantities associated
withclosedcurves encircling the solenoid. Neither is this e¤ectapurelyquantum
one; similarly, in GTR, on parallel transport along a curve encircling a cone,
there is a holonomy e¤ect, depending on the curvature at the tip of the cone,
despite the fact that everywhere along the curve the metric is �at.

4.5 Conclusions

Relationism is commonly thought to take away the reality of space, and in the
context of handedness, to take away the reality of �left�and �right�. But neither
conclusion follows from the invariance principle alone.
In the context of handedness, there are well-known arguments to show

that the concepts �left�and �right�, if they do have meaning, are ine¤able�
incommunicable�and dependent on ostension. Since relationists, rightly, deny
that �left�and �right�di¤er as intrinsic geometric determinations, there is the
temptation to agree that whatever meaning they have is bound up with osten-
sion, and is nonconceptual. But that temptation should be resisted.
The most famous of these arguments is posed by way of a problem, Gard-

ner�s (1964, ch. 18) Ozma problem: �suppose you are in contact with an extra-
galactic civilization, location unknown: to communicate the meaning of �left�
by a language transmitted in the form of pulsating signals�. To this relationists
typically agree the problem is insoluble, because to convey what we mean by
�left�to someone remote from us is to establish extrinsic spatial relations (con-
gruence or anticongruence) between us and them�and that cannot be done by
specifying information only about the relata. This is mixed in with the general
view, traceable to Kant, that �left�and �right�, in practise, are learned by osten-
sion, so cannot be taught from afar, without a common object to which each
party can ostensively refer.
But putting things this way, though not exactly wrong, make facts about

mirroring look more mysterious than they really are. The role of ostension in
language use, essential as it is, brings in questions of much greater scope, that
apply to shape recognition (and much else) quite generally: there need be no
special di¢ culty with the term �left�. And there is a simple solution to the
Ozma problem on the table: the meaning of �left�lies in such-and-such facts
about chiral chemistry, and various relative congruences between enantiomeric
molecules and other dynamically speci�ed bodies. Knowing the meaning of left,
and all there is to know about QET, one knows the relative orientation of every
electroweak decay process. All of this can be conveyed in the equations, without
ever having to know which of the right atlas, or the left atlas, is to be used (for
either can be used).
But do I know, by these methods, which of my hands, �ngers stretched, is

congruent to which dynamically described monochiral molecule? No, not by
these methods�supposing there are in fact no law-like relations between human
anatomy and chiral chemistry. If their relation is purely contingent, nothing
short of observation, ostensive comparison, will do. But given this observation
and the information thus obtained: can I convey it to my distant friends? But

25



of course I can�for it is the information that the normal human heart is on the
side of the hand that is congruent to certain dynamically-speci�ed molecules.
And do they, in that other galaxy, thereupon have to observe objects in common
with us�presumably the same chiral chemistry�to know which hand that is? But
why should observation come into it, supposing they know everything there is
to know about chiral chemistry (as determined by the equations)? They know
which dynamically-speci�ed molecule we are talking about. And they know that
their local chiral chemistry (insofar as it is anchored to the weak interaction) is
the same as ours (for it is the same everywhere)�that is secured by the nature
of �eld theory, as just discussed.
But they may have another question whose answer they want to know �

namely, which of their appendages matches most closely that hand of ours. That
is equivalent to the question, with our signal to them received and understood,
as to which of their appendages is congruent to which monochiral molecule. To
answer that, assuming their gross anatomy is no more constrained by law-like
relations to chiral chemistry than ours, they will have to make observations of
their own. Thus there are two observations, one in each galaxy, each concerning
contingencies of di¤erent anatomies: but neither telling us what �left�really is,
for that has already been dynamically characterized.

Acknowledgments

My thanks to a number of audiences, too numerous to mention, who have heard
me speak on various of these ideas or their precursors, and for helpful discussions
over the years to Harvey Brown, Nick Huggett, Oliver Pooley, David Wallace,
and Graeme Segal, none of whom may entirely agree with my conclusions. My
special thanks to Oliver Pooley for convincing me of the comparative irrelevance
of PCT symmetry to the treatment of mirroring.

References

Bartlett, S., T. Rudolph, and R. Spekkens (2005), �Reference frames, superse-
lection rules, and quantum information�, quant-ph/0610030.
Brown, H. (2005), Physical Relativity. Oxford: Clarendon Press.
Brown, H., and R. Sypel, (1986), �On the meaning of the relativity principle
and other symmetries�, International Studies in the Philosophy of Science 9:
235-53.
Crawford, F., M. Cresti, M. Good, K. Gottstein, E. Lyman, F. Solmitz, M.
Stevenson, and H. Ticko, (1957), �Detection of parity nonconservation in �
decay�, Physical Review 108: 1102-3.
Davis, W. (1970), Classical Fields, Particles, and the Theory of Relativity. New
York: Gordon and Breach.
Earman, J. (1989), World Enough and Space-Time. Cambridge, MA: MIT
Press.
Earman, J., and J. Norton (1987), �What Price Substantivalism? The Hole
Story�, British journal for the Philosophy of Science 38: 515-25.

26



Gardner, M. (1964), The Ambidextrous Universe: Mirror Asymmetry and Time-
Reversed Worlds. New York: Basic Books.
Hegstrom, R., D. Rein, and P. Sanders, (1979), �Parity Non-conserving Energy
Di¤erence between Mirror Image Molecules�, Physics Letters A 71: 499.
Hoefer, C. (2000),�Kant�s Hands and Earman�s Pions: Chirality Arguments for
the Substantival Space�, International Studies in the Philosophy of Science 14:
237-56.
Huggett, N. (1999), Space from Zeno to Einstein: Classic Readings with a Con-
temporary Commentary. Cambridge, M.A: MIT Press.
Huggett, N. (2000), �Re�ections on parity non-conversation�, Philosophy of Sci-
ence 67: 219-41.
Letokhov, V. S. (1975), �On Di¤erence of Energy Levels of Left and Right Mole-
cules Due to Weak Interactions�, Physics Letters A 53: 275.
Mason, S. and G. Tranter (1985), �The Electroweak Origin of Biomolecular
Handedness�, Proceedings Royal Society London A 397: 45-65.
Nozick, R. (2001), Invariances: The Structure of the Objective World. Cam-
bridge, MA: Harvard University Press.
Peskin, M. and D. Schroeder (1995), An Introduction to Quantum Field Theory.
Reading, MA: Addison-Wesley.
Pooley, O. (2003), �Handedness, Parity Violation, and the Reality of Space�,
in K. Brading and E. Castellani (eds.), Symmetries in Physics: Philosophical
Re�ections. Cambridge: Cambridge University Press, 250-280.
Rein, D. (1974), �Some Remarks on Parity Violating E¤ects of Intramolecular
Interactions�, Journal Molecular Evolution 4: 15.
Saunders, S. (2003) �Indiscernibles, General Covariance, and Other Symmetries:
the Case for Non-Reductive Relationalism�, in A. Ashtekar, D. Howard, J. Renn,
S. Sarkar, and A. Shimony, (eds.), Revisiting the Foundations of Relativistic
Physics: Festschrift in Honour of John Stachel. Dordrecht: Kluwer, 151-175.
Stachel, J. (1993), �The Meaning of General Covariance�, in A. Janis, N. Rescher,
and G. Massey, (eds.), Philosophical Problems of the Internal and External
Worlds: Essays Concerning the Philosophy of Adolf Grünbaum. Pittsburgh:
University of Pittsburgh Press.
Ulbricht, T. L. V. (1959), �Asymmetry: The Nonconservation of Parity and
Optical Activity�, Quarterly Reviews (London) 13: 48�60.
Wald, R. (1984), General Relativity. Chicago: Chicago University Press.
Weyl, H. (1952), Symmetries. Princeton: Princeton University Press.
Zel�dovich, B., D. Saakyan, and I. Sobel�man (1977), �Energy Di¤erence between
Right-Hand and Left-Hand Molecules, Due to Parity Nonconservation in Weak
Interactions of Electrons with Nuclei�, Soviet Physics JETP Letters 25: 94-97.

27