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Is symmetry identity?

Marvin Chester

Physics emeritus, UCLA, Los Angeles, California, USA

Abstract Wigner found unreasonable the "effectiveness of mathematics in the natural

sciences". But if the mathematics we use to describe nature is simply a coded expression of

our experience then its effectiveness is quite reasonable. Its effectiveness is built into its

design. We consider group theory, the logic of symmetry. We examine the premise that

symmetry is identity; that group theory encodes our experience of identification. To decide

whether group theory describes the world in such an elemental way we catalogue the detailed

correspondence between elements of the physical world and elements of the formalism.

Providing an unequivocal match between concept and mathematical statement completes the

case. It makes effectiveness appear reasonable. The case that symmetry is identity is a

strong one but it is not complete. The further validation required suggests that unexpected

entities might be describable by the irreducible representations of group theory.

1. Effectiveness Speaks

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In his famous paper entitled, "The Unreasonable Effectiveness of Mathematics in the

Natural Sciences", Eugene Wigner (1979, p. 223) wrote:

"... the enormous usefulness of mathematics in the natural

sciences is something bordering on the mysterious ... there

is no rational explanation for it."

Some think there might be a rational explanation. In his book The Applicability of

Mathematics as a Philosophical Problem, Steiner (1998, p. 5) finds that the use of

mathematics "cannot avoid being an anthropocentric strategy". It rests ultimately on the

human experience of nature. He is thus led to "explore the implication for our view of the

universe" (p. 2) of the evident applicability of mathematics to the physical world. Steiner

turns Wigner's plaint inside out asking what the evident effectiveness of mathematics tells us.

In what follows I will support and expand on this seminal notion.

Wigner was an acknowledged master of group theory, the mathematical theory of

symmetry. Laws of nature - physical laws - are governed by group theory. Bas van Fraassen

demonstrates this in his book, Laws and Symmetry (van Fraassen, 1989). He shows us that

the status of 'physical law' is conferred by symmetry - invariance under the transformations

of nature.

But what is symmetry that it should underlay the very foundation of natural law?

Alternatively put: Why is group theory so effective in describing the physical world? The

answer is that it codifies the basic axioms of the scientific enterprise. The logic of group

theory is the logic of scientific inquiry so that the mathematics we use to describe nature is a

carefully coded expression of our experience.

Group theory is the mathematical formulation of internal consistency in the description

of things. We assume that the system being observed has an intrinsic character independent

of the observer's perspective. It's there. It possesses an objective reality. On this assumption

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- that it's there - how the system is perceived under altered scrutiny must be a matter of

logic. Its appearance follows the logic of intrinsic sameness (Section 7). The codification of

that logic is a matter of group theory. And its success in portraying the physical world is what

validates the assumption.

Rather than ponder its efficacy we take instruction from the mathematics. We know -

by experiment - that the physicist's mathematical description of the physical world is

approximately correct. From the very efficacy of mathematics in describing it we may derive

a message about the nature of the physical world. It is this inversion of Wigner's quest that

we pursue here.

2. How symmetry is identity

I propose that, as used to describe the physical world, symmetry is so elemental that

it coincides with the concept of identity itself. The theory of symmetry is the mathematical

expression of the notion of identification and that is why it is so effective as the basis of

science. By identity is meant the end result of identification, not the other sense of the word

pertaining to identical.

That symmetry has played a substantive role in thinking about nature has a venerable

history dating back to antiquity. This history is nicely outlined by Roche (1987). He discusses

many examples. All of them demonstrate how symmetry considerations have helped solve

physical problems. For example Descartes deduces that "two equal elastic bodies which

collide with equal velocities rebound with these velocities exactly reversed because of

symmetry". (Quote from Roche, 1987, p. 18)

It was Ernst Cassirer who first articulated the idea that the mathematical theory of

symmetry - group theory - may transcend its problem-solving utility. Group theory "has not

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only a mathematical or physical but a universal epistemological interest .." he wrote

(Cassirer, 1945, p. 273). The suggestion was that it has something to do with how we know;

how we evaluate perceptions.

We know objects by their properties. Constitution is what "confers to the carrier of a

set of properties the dignity of an object", says Elena Castellani (1998, p. 182). She

'constitutes' an elementary object of physics - electron, nucleon - from group theory by

showing that invariance under the spatio-temporal transformation group yields as

characterization of the object its energy, its linear momentum, its angular momentum and its

mass. To constitute something, then, is to assign to it labels of significance. The significance

arises from invariance properties.

This exercise exemplifies a broader view of symmetry. Although not explicitly stated

the idea is implied - as it is in the work of others (Van Fraassen, 1989 and Toraldo di Francia,

1998)- that symmetry may be literally equated with identity. To constitute is to identify. We

explore the notion that symmetry is identity; that group theory is the theory of identity.

If, indeed, group theory describes the world in such an elemental way then we must

be able to provide the detailed correspondence between elements of the physical world and

elements of the formalism. This is the way we accept Wigner's challenge to make reasonable

the effectiveness of the mathematics. To display associations between mathematical notions

and physical ones we need concise verbal expressions to match the concise mathematical

ones. Expressions which generate images are the ones that make the mathematics

reasonable.

Using non-technical language, here is a synopsis of the case. The essential quality that

characterizes symmetry is this: the appearance of sameness under altered scrutiny. That this

definition corresponds to group theory as applied to the physical world is grounded in Section

8. But the same phrase - a perceived sameness under altered scrutiny - is just what captures

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the notion of identity. When something is recognizably the same under many perspectives we

grant it identity. An identification is made by labelling. The label tags what it is that we

perceive as invariant. "If there were no invariants we could not define 'identity'" noted M.L.G.

Redhead (1975, p. 78). Here we posit that it is precisely invariance that identifies identity.

Group Theory has an innate taxonomic structure - a taxonomy for behaviour. It

assigns labels for behaviour (appearance) under altered scrutiny. The irreducible

representation labels of Group Theory are the identification markers - the labels of identity.

This is explored in Sections 9, 11 and 12. A central concept in this exposition is altered

scrutiny. It is examined in Section 4. In applications to the physical world it is precisely

altered scrutinies that are the group elements.

To ground the case synopsized in the preceding paragraphs we use a notation that

iconizes the philosophical content: Dirac notation. It is the natural tool for the group theory of

physical processes. What follows is a review of material that every group theory scholar

knows, but recast so as to expose how the mathematics encodes our axioms about how

nature works. To do this we focus on the correspondence between the physical world and the

formalism.

3. The observer, the system, its states and its rules

In examining anything one is an observer. So in asking questions about nature we

focus on what an observer might find. The observer divides the world into distinguishable

parts. Each one, perceived as a discernibly distinct entity, may be called a 'system'. The

system is what is studied by the observer; something accessible to measurement. The

difficulty about the concept lies in the matter of isolation; that the system not be coupled to

the rest of the universe. Evidently such systems are not to be found in nature. It is an

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idealization. Coupling is a matter of degree; never equal to zero. We follow tradition on the

matter assuming coupling weak enough to qualify as zero. 1

An isolated system is always to be found in a 'state'. A state is specified by its

appearance by which is meant the results of a compatible set of measurements on the

system. According to quantum mechanics these are eigenvalues of each of a complete set of

commuting operators. 2 This is discussed further in Section 9. What is relevant here is that by

'appearance' is meant a set of numbers marking compatible measurement results. We label

that set of numbers by a single integer, n . A state of the system is designated by n , using

standard Dirac notation.3

A system is governed by a rule. Its behaviour is determined by a law - or rule -

through which its nature is expressed. In physics this takes the form of a Hamiltonian a

Lagrangian or a variational principle.

The observer makes the measurements. He or she is equipped with a battery of

instruments. The observer's eyes are one such instrument but he or she will usually have

others - like clocks, meter sticks, electrometers, particle detectors, etc. In accordance with

von Neumann's principle of Psychophysical Parallelism, it doesn't make any difference

whether the instrument is within the observer's body or not. (See von Neumann, 1955, p.

419-420) He or she scrutinizes the system by recording the measurement results his

instruments read. Making sure that all his measurements are compatible he or she assigns to

a particular collection of his or her measurement results a particular value of n . He or she

concludes that the system is in the state n .

4. Altered scrutiny

The thread by which symmetry and identity are bound together is altered scrutiny. An

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altered scrutiny of the system means that the observer moves to another frame of reference

to record his measurements. To this new reference frame he takes with him his entire

ensemble of instrumentation - as if they were, indeed, a part of the observer's body. By

altered scrutiny is meant the action that puts the observer into position to make his new

measurements. It excludes the act of measurement itself - the scrutiny. The latter can

change the state of the system. The former cannot.

The formal mathematical term for altered scrutiny is transformation. If what one

means by transformation is altered scrutiny then the connection between identity and

symmetry becomes clear. It is the observer's altered scrutiny that transforms the state.

Altered scrutiny is purely an observer's construct. It is a reweaving of the fabric of descriptive

space, an enterprise that goes on in the mind of the observer.

The prototypical change in reference frame is a physical rotation of the observer's

coordinates. Figure 1 schematizes the idea. There the system's state is portrayed as a

function in some descriptive space in which a representative vector is ! ! . To each ! !

corresponds an amplitude

n

! !

which measures the strength of the state at the point ! ! . In

the figure the function

n

! !

is the pyramid whose projection we see.

The key feature of altered scrutiny is this: that corresponding to the observer's action

of altering her scrutiny there is an operator Gg whose effect is to produce a new state,

Gn

g

, from the state n , observed originally. The new state, Gn

g

, is called the

transformed state. And Gg is the transformation operator. This is illustrated in Figure 1.

What Gg produces when it acts on n , we have called a 'state'. This proclaims a notion

about reality: that altering one's perspective on a system cannot change its intrinsic nature.

We see the same system, but in some other state. Its appearance is altered.

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Whatever represents a physical system has the technical property that its states

constitute an invariant4 vector space for the group of all altered scrutinies. This is the formal

way of saying that from a state of the system an altered scrutiny can only produce another

state of the same system. We don't alter the system by looking at it. That is the significance

of calling Gn

g

a 'state'. Mathematics allows other possibilities.5 Not all vector spaces are

invariant to group elements. So our notion about reality is expressed as a mathematical

constraint. Its consequences are pursued in equation (4) of Section 10.

We enumerate altered scrutinies by subscripts. The gth altered scrutiny is Gg. The

figure shows one of the altered scrutinies belonging to the continuous Lie group, SO(2). In

that case we can use θ for g. The altered scrutiny displayed is GG

g

!

=

or

( )

G ! .

That altered scrutinies fall into groups follows from the definition. They conform to the

five defining properties of a group. 1. Their law of combination is simply sequential action.

Two alterations in scrutiny performed one after the other is written in product form, G Ga

b

transformed

state

original

state

θ

φ

η

x

y

θ

φ

transformed

observer

original

observer

ξ

r

ρ ρ

Figure 1. The observer's altered scrutiny is what transforms the state.

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where the order means Ga is done first. 2. Closure is guaranteed because two alterations in

scrutiny performed sequentially amounts itself to an altered scrutiny. The observer reorients

himself and then, without taking measurements, does so again. He could just as well do it in

one step. 3. Associativity obtains simply because there is no meaning to inserting

parentheses among a product of altered scrutinies. They are simply a series of sequential

actions. 4. All inverses exist. Simply undo the alteration in scrutiny. And 5. The identity

element is the 'do nothing' action. Don't alter scrutiny. Thus altered scrutinies can always be

grouped.

Altered scrutiny or transformation is something that produces a mapping of the states

of the system onto themselves. In this generality it is not easily conceivable how an observer

might physically execute many altered scrutinies. Inversion is an example. It may not be

physically executable but it is conceptually executable: replace position variables by minus

themselves. The observer alters his description of the system. That is his altered scrutiny.

The phrase generates an image in the mind: those transformations that an observer can

physically execute. We wish, in the definition, to embrace abstract altered scrutinies;

transformations that one may not be able to execute physically but that one can imagine -

that one can execute mathematically - including, say, changes in motion and even of phase

in a wave function.

5. The system remains inviolate

Altered scrutiny actions have nothing to do with the system. They have only to do with

the observer. The observer uses the same measuring instruments to make his

measurements, but from a different perspective. In texts on group theory the distinction is

made between the passive view and the active one.6 They differ only in labeling. Aside from

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this they are mathematically equivalent. They are not equivalent philosophically.

What is traditionally called the passive view is shown at the left of Figure 1. The

observer moves. The system is left inviolate. The active view is described as altering the

physical system under inspection.7 That is how one might interpret the right side of Figure 1.

It is an interpretation we deliberately avoid. The right side of Figure 1 results - not from

altering the system - but from altering the observer. The system's appearance is altered - but

its integrity remains untouched. It is transformed but always left undisturbed.

The equivalence of the views mathematically assures us that in the use of group

theory to describe a physical system we may always ascribe the transformation to a change

in the observer's perspective. An example is the particle exchange transformation. It may be

entirely reinterpreted as a relabeling of particles by the observer. (Fonda and Ghirardi, 1970,

p.39) In this view the exchange is not of particles but of the enumeration labels used by the

observer to tag them.

6. Descriptive space

When the observer alters his perspective he is readjusting his basis in descriptive

space. In the figure his new basis is (x,y), his old is (ξ,η). The altered scrutiny action, Gg, is

what produces the new basis from the old. If the physical point at position ! ! is described

under altered scrutiny as being at r then the meaning of Gg is embodied in the expression

Gg=

! !

r

. By Gg is meant, here, the prescription for delivering the appropriate r for the

given ! ! . In that way Gg implements the altered scrutiny. This prescription, for the case in

the figure, is best delivered in matrix form.

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x

y

!

####

"

$

%

&&&& &=G(!)"

#

!

####

"

$

%

&&&& &=

cos!

'sin!

"

sin!

cos!

!

####

$

%

&&&& &

"

#

!

####

"

$

%

&&&& & (1)

This is a compact rendering of x=x(ξ,η) and y=y(ξ,η), the equations represented by

Gg=

! !

r

.

An observer can reverse his scrutiny alteration thus retrieving ρ ρ from r. Put

symbolically

1

Gg!

! ! = =

r . So a state of the system may equally well be portrayed in the

transformed coordinates, r, by just carrying out the coordinate transformation; substituting

1

Gg!r for ! ! . The mathematical rendering of this statement is

1

nGn

g!

=

! !

r

(2)

This notates the idea that altered scrutiny is a matter of rewriting the descriptive space basis

in terms of the new basis coordinates.

7. The principle of intrinsic sameness

We now have two different prescriptions for the operator, Gg, that represents an

alteration of an observer's perspective. It produces a transformed state, Gn

g

, from the

original n in Section 4. It generates a new descriptive space, r, from the old one,

1

Gg!r in

Section 6. That the two prescriptions be commensurate represents a philosophical

commitment: that the altered appearance of a system (its transformed state) is due only to

the altered scrutiny of the observer (the transformed coordinate system) and to nothing else.

That is the content of Equation (3).

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1

GnGn

gg

!

=

rr

(3)

The left hand side gives the state's amplitude at a certain point of descriptive space.

The right hand side insures that the transformed state has the same amplitude at the same

physical point in that space. It ordains that the transformed state Gn

g

be what arises from

altered perspective. It does this by giving us the prescription for discovering it

computationally. Equation (3) is just that prescription. It is the mathematical formulation of

our elemental intuition that a system has properties independent of our scrutiny; that even

though the system appears different under altered scrutiny its intrinsic sameness is

preserved. Thus, equation (3) expresses the logic of intrinsic sameness; that the system is

there regardless of our scrutiny of it. The application of group theory to the physical world -

to physics - is based on this equation. In texts it goes by the formal name of "induced

transformation".8 But this term fails to capture its formidable philosophical significance.

8. Symmetry is apparent sameness under altered scrutiny

We are now prepared to examine what it is that we mean by symmetry. What is the

essential test by which we decide operationally that something is symmetric? Because the

word is in everyone's vocabulary, a meaning in 'laymen's terms' often dissolves into a

plethora of examples. And a definition becomes a matter of abstract mathematics.9 We wish

to define it in laymen's terms, succinctly but with a view to accommodating the mathematics

because it is through simplicity and suggestive imagery that the mathematics appears

reasonable.

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Symmetry is apparent sameness under altered scrutiny. This phrase certainly captures

the experience of our visual sense of symmetry. Visually any object that we call symmetrical

has this key property: that under some altered scrutiny it looks the same to an observer. It is

congruent to itself. In most cases, appearance changes when we alter our scrutiny. Among all

the altered scrutinies possible only a few produce same appearance. Those few define the

symmetry. But 'same appearance' need not be restricted to the visual. 'Same measurement

results' qualify also. The definition is an operational one; scrutinize to check for sameness.

It's important to distinguish between intrinsic sameness and apparent sameness.

Intrinsic sameness is always preserved. What constitutes symmetry is that an observer's

altered scrutiny leave the object's appearance unchanged. When apparent sameness is

preserved under some altered scrutiny, symmetry is present. The technical expression,

'invariant under the transformation' is what 'perceived sameness under altered scrutiny'

means.

If there are some altered scrutinies then there are a group of them. We generate the

group by applying already discovered same-appearance-scrutinies in sequence until all the

members of the group emerge. Altered scrutinies are the group elements of group theory.

These same-appearance-scrutinies refer, of course, to what are formally called 'covering

operations' or 'symmetry operations'. Each same-appearance-scrutiny is the inverse of a

symmetry operation. They differ in point of view: one moves the system, the other the

observer.

9. Measurements provide enumeration labels

The states of a physical system are characterized by constellations of measurement

results: the ones enumerated by n. As archetype we think of the spectrum of deflections (the

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states) of an atom beam (the system) passing through a magnetic field gradient as in a

Stern-Gerlach experiment. (Chester, 1987, p. 148-150) Physical states correspond to rays in

Hilbert space. Each state, n , is a basis vector in this space. Thus Hilbert space is

constructed from measurement results. What the observer sees - via his scrutiny apparatus -

are a set of basis states in the Hilbert space of the system.10 Any state in the space is a

superposition of the basis states.

10. Altered scrutinies generate group representations

What is perceived under altered scrutiny, Gm

g

, is some state of the system. It is

within the system's Hilbert space. Since any state of the system is a superposition of basis

states so must be the altered scrutiny state. Put mathematically the statement is:

rGgm =

r nn Ggm m = any of the n

n

!

(4)

Formally this equation says that the Hilbert space with basis, n , is invariant with respect to

the group {Gg}. Physically it expresses the reasonable expectation that from a state of the

system an altered scrutiny can produce only another state of the system. Because the

observer carries his descriptive space with him, any state is describable in terms of his basis

states,

n

r

, even the transformed state,

Gm

g

r

, that he encounters because of his

alteration in scrutiny.

The coefficients in the sum - written n Gm

g

- represent the extent to which the

transformed state Gm

g

is like n . They are deducible computationally by incorporating

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the intrinsic sameness rule of equation (3) into equation (4). Choosing to write these

coefficients as n Gm

g

, exposes their nature: elements in a square matrix. The matrix is

called a representation of the transformation. It is a mathematical rendering of the particular

altered scrutiny action, Gg. When gathered for a group of such altered scrutinies the

matrices of coefficients constitute a representation of that group. That is because

multiplication of the corresponding matrices implements the sequential performance of

altered scrutinies.

But these coefficients are connected to the statistics gathered in observations. They

are experimentally accessible. The relative number of times an altered scrutiny of the state

m will unearth the state n is just

2

n Gm

g

. So the group representation matrix

elements, n Gm

g

, are associated with measurement results.

An observer examining the states of a system cannot help but generate a matrix

representation of the group as she goes through her group of altered scrutinies. The matrices

in the representation will always be square and invertible. Invertible because an altered

scrutiny always has an inverse. Square because both n and m belong to the same basis of

states. A fundamental notion in group theory is that its mathematics is completely described

in terms of invertible square matrices. Just such matrices arise naturally from examining the

physical world.

11. Irreducible representations yield labels

The important thing about a matrix representation of a group is that it can be reduced

to irreducible representations - to a direct sum of special symmetry matrices. These embody