Philosophy of Science, 69 (September 2002) pp. S221-S234. 0031-8248/2002/69supp-0020
Copyright 2002 by The Philosophy of Science Association. All rights reserved.
Gauge Principles, Gauge Arguments and the Logic of
Nature
Christopher A. Martin
Indiana University
http://www.journals.uchicago.edu/PHILSCI/journal/issues/v69nS3/693020/693020.
html
I consider the question of how literally one can construe the "gauge argument," which is the
canonical means of understanding the putatively central import of local gauge symmetry principles
for fundamental physics. As I argue, the gauge argument must be afforded (at best) a heuristic
reading. Claims to the effect that the argument reflects a deep "logic of nature" must, for numerous
reasons I discuss, be taken with a grain of salt.
Symmetry principles have moved to a new level of importance in this century and
especially in the last few decades: there are symmetry principles that dictate the
very existence of all the known forces of nature.
S. Weinberg
Send requests for reprints to the author, 130 Goodbody Hall, Bloomington IN 47405;
[email protected]
I thank Jon Bain, Tony Duncan, John Earman and Chris Smeenk for their comments on an earlier
draft.
1. Introduction.
This sentiment expressed in the above quotation is characteristic of the
canonical understanding of the structure of fundamental physical theory today. The
symmetry principles that have come to prominence in recent decades are so-called
local symmetry principles. Perhaps the most common way of presenting and
understanding the workings or content of these symmetry principles is in the
context of "the gauge argument," a fixture in the physics literature. Briefly, this
argument starts from some known non-interacting charged matter field with a global
symmetry and, through demanding that symmetry be made local, dictates the
introduction of a new gauge field interacting in a prescribed way with the original
matter field. The argument is taken to capture the operation of a fundamental
physical principle, "the gauge principle": the principle holding that the form of the
fundamental interactions is to be determined, or dictated, via requirements of local
(gauge) symmetry. For the class of interacting field theories resulting in this way
from the demand of local invariance, gauge field theories, contains those theories
accurately describing what we find in nature! Given this, it might be hoped that the
operations of the gauge argument and/or its purported universal applicability to all
fundamental interactions, would provide the key to a proper understanding of the
nature of gauge invariance and of the gauge structure of fundamental physical
theory.
Read literally, the gauge argument might appear to have an air of magic about it,
to effect a sort of "epistemic miracle." For one seemingly gets a fair amount of
"physical output" from but a modicum of "physical input." Faced with the argument
presented in this schematic way, it is difficult not to be awed by what is apparently
its sheer physical fecundity. However, the argument presented in this way seems to
me just too good to be true, and indeed I claim it is. As the saying goes, "you don't
get something for nothing," and not surprisingly the argument when looked at more
critically is not quite so magical.
In this paper, I warn against reading the gauge argument too literally in this way,
as a sort of laying bare of the "logic of nature." There are three points at which a
literal reading of the gauge argument falters. (1) The received wisdom concerning
motivations for key parts of the argument is suspect, and (2) the argument does not
proceed unambiguously. As I discuss in section 3, even if the initial requirement of
local (as opposed to global) gauge invariance can be justified in some natural, nonquestion-begging way it is not clear that it can be this requirement does not
alone uniquely dictate the form of the full theory. Moreover, (3) there are wholely
other approaches to the logic of nature, ones that in effect assign a different origin
to the gauge structure of fundamental theory. As I discuss in section 4,
considerations surrounding renormalizability and effective field theories, for
example, might be taken to supplant, indeed obviate, any principle of local gauge
invariance as responsible for the "shape," including the gauge structure, of physical
theory. Such considerations raise the interesting possibility that there is less to any
fundamental "gauge principle" than the canonical understanding takes there to be.
2. The Gauge Argument.
The canonical way of understanding the workings or content of local gauge
symmetry principles is by pointing to the algorithm for producing interacting field
theories from the demand of local gauge invariance
the "gauge argument."1 Given
the success of the gauge paradigm, it playing a key role in the developments
culminating in the most far-reaching and successful physical theories we have ever
known, some take the argument to represent the very logic of nature. Symmetry
principles as embodied in the argument are then taken to express/reflect deep
features of the physical world.2 In order to make some remarks concerning how
literally we can construe the operations of the argument, we first look at a concrete
example.
Consider a field representing electrically charged matter. The free field obeys
the Dirac equation which is just the Euler-Lagrange equation(s) for the Lagrangian
(density)
Dirac
= (i
- m) .3 The corresponding action is clearly invariant under
so-called "global" U(1) phase transformations:
eiq
;
e-iq with a constant.4
It follows from Noether's first theorem that when the equations of motion are
satisfied there will be a corresponding conserved current.
Consider now "localizing" these phase transformations, i.e., letting become a
function (x):
eiq
(x)
;
e-iq
(x)
. The free field Lagrangian is clearly not
invariant under such transformations since the in general non-vanishing derivatives
of the arbitrary functions, i.e.,
(x), will now appear in the transformed
Lagrangian. In order that the field admit the local transformations as variational
symmetries, the Lagrangian must then be modified. In particular, we replace the
free field Lagrangian with
=q
interacting
= (i
- m) - qA
Dirac -
J A , with J
. The current J is in fact just the conserved Noether current associated
with the global U(1) invariance. Towards securing local invariance we have
introduced the field A , the gauge potential. The particular form of coupling of the
matter field to this gauge potential in
interacting
is termed minimal coupling.
This modified Lagrangian is now invariant under the local phase transformations
provided that the vector field A is simultaneously transformed according to A (x)
A (x) -
(x). This transformation behavior is, of course, familiar as the
covariant analog of the well-known electromagnetic gauge transformations. This
suggests the possibility of viewing the new field A as representing the
electromagnetic potential.
Pursuing the idea of viewing the field A as representing the electromagnetic
potential, we note that the Lagrangian
interacting
does not yield a fully interacting
theory. Varying the Lagrangian with respect to the matter fields yields the latter's
equations of motion
the field now being coupled to the A field. But, it remains to
add a "kinetic term" for the A field itself. Such a kinetic term, in effect, imbues the
vector field with its own existence, accounting for the presence of non-zero
electromagnetic fields, for the propagation of free photons. The Lagrangian
kinetic =
Maxwell
=-
F
F
, with the gauge field F
defined as F
:=
A -
A gives
(source-free) Maxwell's equations. Putting this all together yields the Lagrangian for
the fully interacting theory:
Maxwell-Dirac
=
interacting
+
kinetic =
Dirac
-J A -
Maxwell.
The inhomogeneous coupled equations of motion for the gauge field (the
electromagnetic field) now follow from varying the full action with respect to A .5
Finally, consider that a mass term for the vector field A of standard form
mass
=
photon-
m2A A . is not gauge invariant. In keeping with the demand of local gauge
invariance, the vector field A (i.e., the photon) must then be massless. Local
gauge invariance thus necessitates a massless photon.6,7
3. Does Local Gauge Symmetry Dictate Interaction? Demystifying the
Gauge Argument.
Having provided a sketch of how the gauge argument works in a concrete
example, let us proceed to consider how literally the argument can be construed.
Does the argument truly dictate the introduction of new physics in a simple and
direct manner? Not surprisingly, the gauge argument cannot proceed as simply as
this schematic presentation might make it seem.
3.1. From Whence the Local Gauge Invariance?
Most immediately what are we to make of the initial, central demand of local
gauge invariance? The demand is anything but self-evident and presumably, in the
context of the gauge argument, must be argued for on some basis. Unlike the
global invariance, the demand for the corresponding local invariance does not have
an immediate physical counterpart.8 Is it to be taken as a direct implementation of
some sort of unassailable first principle? If so, is the demand (or principle)
something with which we are already familiar only in a different form?
A common justification for the demand of local gauge invariance in presenting
the gauge argument is to present it as some sort of "locality" requirement. In
outline, the "gauge locality argument" is that global gauge invariance is somehow at
odds with the idea of a local field theory, and that to remedy this we must instead
require local gauge invariance. This rather brief argument is just how Yang and
Mills motivated the demand in their seminal 1954 paper, very much setting the tone
for subsequent treatments.9 Just what to make of this argument is not clear,
however, there are many interrelated senses of locality that might be at issue.
At a general level, making precise the connection between the global/local
distinction as it figures in classifying gauge transformations and in the spacetime
sense is non-trivial. The fields and transformations ostensibly "live" in configuration
space and need to be "brought down" to spacetime.10 Also, the local versus global
distinction for gauge transformations as commonly employed is itself in general
dependent on a choice of gauge this stems from the fact that in order to construe
the fields (and transformations) as living in spacetime one must choose a gauge in
the first place. This is made clear in treatments of gauge field theories in the
principle fiber bundle framework, where gauge transformations are bundle
automorphisms and the typical global/local distinction for gauge transformations,
being gauge dependent, is seen to be of little use (Bleecker 1981).
Generally, a "local" field theory is, according to the canonical understanding, just
one specified by a Lagrangian depending on the values of the field and a finite
number of its derivatives at a single point.11 This is understood to capture the idea
that the equations of motion
partial differential equations determined from this
Lagrangian through a variational procedure should be such as to determine the
behavior of the field at some point only in terms of what is happening at and around
that point. The "locality" at issue here cannot be used to motivate the requirement
of local gauge invariance over and above global invariance since it clearly does not
exclude global symmetries. We have already seen that a field theory can be local in
this sense and yet have a global gauge invariance. The free Dirac field described by
the Lagrangian
Dirac
depends on the field and a finite number of its derivatives at
a single point and thus is a "local" field theory in the above sense. Yet this
Lagrangian is invariant under the group of global U(1) gauge transformations.
Perhaps we could take the sense of "locality" at issue in the gauge argument to
be tied more specifically to the dictates of special relativity (STR). Along these lines,
the demand of local gauge invariance might be presented as providing a sort of "fix"
for a global gauge invariance taken to be somehow at odds with the heart and spirit
of STR.12 Let us call this particular form of the locality argument the "STR gauge
locality argument." The claim/argument is presumably that the group of local gauge
transformations
loc
and not the group of global gauge transformation
glob
are
compatible with the dictates of STR. If this position could be maintained, then we
would apparently have a clear physical basis for the demand of local gauge
invariance and, with that, an underwriting for the gauge argument.
One way of understanding the demands of STR in this context might be to take it
as a prohibition of superluminal communication.13 Thinking of the demands of STR
along these lines, one might argue that to perform a global gauge transformation
would in effect be to perform "the same" transformation at each and every point,
and that this seems to require/enable superluminal communication. But, I take it
that this argument rests on blurring different senses of locality. It blurs matters of
the (light-cone) structure of Minkowski spacetime in STR with that distinguishing
from
glob.
loc
And, it is clear that the former cannot be of direct significance to
motivating the latter. For suppose that in the spirit of the "STR gauge locality
argument," we eschew symmetry transformations wherein the same transformation
is performed at points not in causal contact. Whatever its details, it is clear that the
set of transformations compatible with STR along such lines,
STR,
is not the class
of transformations which figures in and is crucial to the formulation of gauge
theories,
loc.
In fact, the latter class of transformations contains as a subset (more
precisely, as a rigid sub group) just those transformations
gauge locality argument aims to exclude.
glob
which the STR
In any event, it would seem that such a local-gauge-as-locality argument
(whether of the specific STR variety or not), if it can be mounted, must rely on an
active reading of the local gauge transformations. The force of any such argument,
presumably, derives from the untenability of performing the "same" transformation
everywhere at the same time (i.e., a global transformation), this violating some
notion of locality. Yet if, as the received view holds, the transformations are viewed
as merely passive coordinate relabelings
even " ultra-passive" in the sense that
there is no actual physical change of a physical reference frame but only a change
in purely mathematical formal apparatus then it seems that global transformations
cannot possibly pose any threat to locality. This because, by stipulation, there is
nothing physical that gets changed under the transformations. Thus, I do not see
how one can mount any argument for local gauge symmetry in the name of locality
if one ascribes to the received view of gauge symmetry/invariance.14
I have considered only a (related) few of the most common ways of motivating
the initial demand for local gauge invariance in running the gauge argument. For
the remainder of the paper, I take it on faith that the demand can be justified in
some non-question-begging way. This faith seems not too unreasonable given that
the ascription of deep physical significance to local symmetry principles has come
to such prominence in recent decades. The appeal to other elements figuring so
deeply in the construction and/or structure of physical theories
Lorentz invariance
for example,
has been well rationalized. Surely this one has too! On this
faith (shaky as it may in fact be), let us continue to consider the prospects for a
literal construal of the gauge argument.
3.2. "Fixing" the Lagrangian Guidance from Gauge Invariance.
As we saw in section 2 the demand for local invariance of the action for the free
field meant that one must necessarily modify the Lagrangian. This stems from the
fact that sensible Lagrangians involve derivatives of the physical fields. These
derivatives, though, do not transform covariantly (i.e., like the fields themselves)
under local gauge transformations since the derivatives of the (arbitrary) functions
specifying the local transformations enter. Thus one must modify the Lagrangian in
order to cancel out these extra terms, and this modification includes the introduction
of a new field, the gauge potential, coupled to the now interacting matter field.
One important issue that I (purposely) glossed over earlier was the matter of the
uniqueness of the modified Lagrangian under the imposition of local invariance. If
we hope to make any strong sense of the gauge argument as truly "dictating" the
form of the theory the correct theory we expect that the locally gauge invariant
Lagrangian be unique. Not surprisingly, it is more complicated than this.
First, if producing a gauge invariant action or Lagrangian were our only guide
then there are clearly many other gauge invariant terms that could be added to the
Lagrangian. For example, in addition to the "minimal term" qA
which was
added above, we could have added, say, higher order covariant derivatives, e.g.,
or
F
, etc. Or, one might include a dependence of
interacting
on F
. More
interestingly, one might consider the addition of a Pauli term, one proportional to [
,
] F
, which is both gauge invariant and Lorentz invariant. Such a term would
in fact have direct physical consequences making the magnetic moment of the
electron an adjustable parameter (Weinberg 1995, 517). Such terms are thus
apparently not included in the gauge invariant Lagrangian correctly describing
nature.15
In order to pick out the minimal modification uniquely, we must bring in besides
our general knowledge of field theories e.g., that only a certain number of
derivatives should appear in any term of the Lagrangian if the equations of motion
are to be of, say, second order in the derivatives of the fields the requirements of
Lorentz invariance, gauge invariance, simplicity and, importantly, renormalizability.
The minimal modification is then the simplest, renormalizable, Lorentz and gauge
invariant Lagrangian yielding second order equations of motion for the coupled
system (O'Raifeartaigh 1979). The key point is that, in the context of the gauge
argument, the requirement of local gauge invariance gets a lot of its bite in
combination with other formal and physical requirements.
One might argue that, at the least, other things held fixed some requirement of
formal simplicity selects the minimal modification as the unique gauge invariant
modification. In this way, the demand of local gauge invariance might be equated
with a principle of simplicity. While assumptions of simplicity have certainly proven
valuable guides in past theorizing, there is no reason, though, to think that they
provide unambiguous, let alone infallible guides in constructing theories and/or in
construing any logic of nature.16 I suspect that these remarks are likely to little
phase a good physicist, who would claim that the argument requires completion
rather than critique. My point has been only that the demand of local gauge
invariance is not the sole input to the gauge argument nor is it necessarily the most
important.
Setting aside this matter of the uniqueness of the gauge invariant minimal
coupling, another important point is that, in contrast with how it is often portrayed,
one does not strictly speaking "generate" a new interaction field in running the
gauge argument. This gauge field, insofar as it is a physical field, is ultimately put in
by hand. The gauge potentials (in our example, A ) generated in the gauge
argument form a restricted class of all such A fields: since we start with a free
matter field they are of course all gauge transformable to the zero field. Such
potentials, though, correspond to zero F
fields.17 Thus one generates only
physically trivial gauge fields in running the gauge argument. It is thus not clear why
in the context of the gauge argument one would necessarily add the kinetic term
Maxwell for
this gauge field to the Lagrangian. For it is this addition and the
subsequent varying of the action with respect to the gauge potentials that "gives
physical life" to the field.
In the end, an important physical generalization is made in adding the kinetic
term (by hand) to the Lagrangian. The generalization is from a non-physical, formal
coupling of the matter field to trivial gauge fields (since F
0) to the physical
coupling of the matter field to non-trivial gauge fields (F
0).18 In making this
generalization, one puts in by hand much of the important physics of the new
interacting theory.19 This point goes a long way toward explaining the easily
acquired illusion of getting more physics out of the gauge argument than one puts
in.
The most I think we can safely say is that the form of the dynamics characteristic
of successful physical (gauge) theories is suggested through running the gauge
argument. This is not to say that the requirement of local gauge invariance cannot
serve as a useful selection criterion for generating possible modifications of a free
Lagrangian including the addition of (what we interpret as) a kinetic term associated
with any newly introduced interaction field(s). It can and does serve such a purpose.
In fact, historically, there was just such a pragmatic appeal to local symmetry
principles.20 In any event, it is not how a straight-forward, literal reading of the
gauge argument portrays it: i.e., it is not the case that by itself the demand of local
gauge invariance (1) dictates uniquely the form of the interacting theory or (2)
strictly speaking dictates the existence of, or accounts for, the origin of a new
physical gauge field. In order to pick out the correct form of the theory, other
considerations must ultimately enter. And, it is not at all clear that these other
considerations or requirements are in any sense inferior conceptually or physically
to that of local gauge invariance.
In fact, quite the opposite. In lieu of any strong arguments for a physically based
and uniquely implemented principle of local gauge invariance, perhaps the best
chance we have of arguing for the uniqueness of the interacting Lagrangian
Dirac
Maxwell-
is from a direction stressing certain of these other considerations. As we
discussed, the demand of local invariance must be supplemented at various points
of the gauge argument toward arriving at the fully interacting gauge theory correctly
describing nature. Besides the prominent figuring of our prior knowledge of physical
field theories, there are other fundamental requirements or constraints on theory
such as Lorentz invariance and renormalizability. This last requirement in particular
is a powerful one. For instance, a Pauli term, which is not excluded by the
requirements of Lorentz invariance and local gauge invariance, and whose addition
to the Lagrangian would drastically change the theory, is non-renormalizable and
can be excluded on that basis. The next section pursues the idea that
renormalization and/or other, even more fundamental, considerations, might bear a
larger burden for the structure in particular, the gauge structure of our
successful theories than does the operation of any fundamental gauge symmetry
principle.
4. Turning it Around: Other Approaches to Gauge Invariance.
Up to this point, I have followed the well-traveled path of treating the demand of
local gauge invariance of the theory as part of the "physical input," the purportedly
principled physical starting point, and the associated full-blown (gauge field) theory
describing interactions as the "physical output." This way of viewing the gauge
structure of fundamental theory is of course precisely that suggested by the
canonical view of the gauge argument as the embodiment of a deep physical gauge
principle. However, there are other ways of thinking about why our theories are the
way they are, so to speak, and some of these have gauge invariance as more of an
"output" than an "input." For example, there are arguments to the effect that various
consistency requirements, mathematical and/or physical, require theories of, for
example, self-interacting spin one particles to be of Yang-Mills form with its
characteristic group properties.21 Such arguments clearly paint the gauge
invariance of physical theory in a different light than does the canonical view.
A more prominent approach which "in effect" turns the received view on its head
is that of placing renormalizability (or, alternatively, perturbative unitarity) at the
base of fundamental theory. It can be shown that the requirement of
renormalizability (resp., perturbative unitarity) requires that theories have the
characteristic form of (spontaneously broken) Yang Mills gauge theories.22
Renormalizability can be tied directly to a theory's being well behaved in the sense
that it make sensible predictions for quantities of direct physical interest. Thus, one
could reasonably argue that gauge invariance is but a feature of the class of wellbehaved (renormalizable) theories that happen to correctly describe the physics at
hand.
Interestingly, renormalizability has itself arguably been superseded in a certain
sense. According to the currently prominent effective field theory program, the
familiar renormalizable (quantum) field theories are actually low energy
approximations to some more fundamental underlying theory. Besides the finite
number of familiar renormalizable terms, such effective theories (rather, the
actions) necessarily contain an infinite number of non-renormalizable terms. But as
long as at high energies there is some underlying well defined theory (e.g., strings,
loops, etc.), then at much lower energies these non-renormalizable interactions will
be highly suppressed and thus will be calculationally insignificant, though not
physically absent.23 That is, the low-energy "residue" will in fact "look like" a
renormalizable theory.
This, so the thinking goes, explains why we have gotten by so well focusing on
the renormalizable terms. And, as I mentioned above, the renormalizable theories
will necessarily be (spontaneously broken) Yang-Mills gauge theories. Such a view
changes once again the whole picture with regard to what we take to be
"fundamental." Given consideration of effective field theories, the appeal of
ascribing any deeply fundamental significance to the gauge structure of our
theories, especially as resulting from the operations of some deep physical gauge
principle, is further diminished.24 This gauge structure, it could reasonably be
argued, is but a direct consequence of (1) the empirical fact that there exist
interacting spin one particles in nature and (2) the assumption that our theory of
such particles is the residue of some more fundamental underlying theory (e.g.,
string theory). Under these assumptions, we arrive at the familiar Yang-Mills gauge
theories describing such particles. So, we might avoid altogether any appeal to
gauge symmetry principles in describing the shape or content of current theory.
Gauge invariance arguably becomes an incidental, albeit interesting feature of
successful renormalizable quantum field theories.
The alternative approaches to the logic of nature discussed in this section call
into question the very idea of a true physical "gauge principle." How, given this,
might we account for the appeal of the more prevalent view of gauge invariance as
just such a fundamental physical principle? There is of course the obvious answer
that considerations of renormalizability and effective field theories is a relatively
new business, and that historically there was in fact a firm heuristic basis for appeal
to local symmetry principles.25 This is not to slight the significance of local gauge
symmetry principles but rather to call attention to their true historical role. The
history and the nature of the relevant physics seems to call out for little more than a
heuristic reading of the gauge argument and of the role of gauge symmetry
principles.
Why, though, might flowery language concerning all-powerful gauge symmetry
principles persist despite their contrasting and respectable historical role as
heuristic guides? I think that a reasonable answer to this question might run as
follows. Gauge invariance is indeed a very conspicuous, even powerful feature or
property of theory: conspicuous and powerful just in that, formally, this feature is
directly related to many characteristic properties of direct interest. To name but a
few: the fact that the massless limit of the (quantum) propagator for vector particles
is smooth, the fact that intricate cancellations occurring in certain calculations make
the theories renormalizable, or even the fact that gauge theories can be given a
geometrical formulation in terms of principle fiber bundles. The pitfall to be avoided,
however, is that as a feature of theory so rich in interesting affiliations, it is perhaps
too easy to uncritically elevate this feature to the defining "principle" of the theory,
the one from which all, including the form of the theory itself, is to be derived.
Indeed, this gauge invariance is related to many interesting features of the theories
in which it figures. But, it does not necessarily follow that we should be so awed by
these interesting relationships that we exalt them in the form of an invented physical
gauge principle operating in nature.26 This is made especially clear in light of the
considerations just discussed which take the associated gauge structure to be a
"side-effect" of working with the low-energy approximations ("residues") of some
more fundamental underlying physical theory.
Conclusion.
In this paper I have discussed three key points at which a literal reading of the
gauge argument runs into difficulties. Consequently, the claim that the argument, as
the embodiment of some fundamental physical gauge symmetry principle,
"dictates" or "determines" the form of fundamental interactions as well as the
existence of certain physical fields must be taken with a large grain of salt. The
motivations for the demand of local gauge invariance in the first place are not
unambiguous the burden remains to justify this demand in some non-questionbegging way. Moreover, the choice of modified Lagrangian given just this demand
is not unique, and other fundamental concerns must enter toward specifying the
correct physical Lagrangian. And, the claim to "generate" a new gauge field in
running the argument is not quite right. The imbuing of this new field with its own
substantial reality, as a physical, coupled gauge field, is, once again, something
that is not dictated by the local gauge invariance requirement per se, and must be
done by hand. Perhaps nothing calls more into question a literal reading of the
gauge argument than does the existence of quite different approaches to the gauge
structure of our fundamental theories, ones which have the "logic of nature" wholely
different. Such approaches indeed raise the possibility that, strictly speaking, there
is nothing deserving to be called a physical gauge principle at work in fundamental
physics.
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What I here call the "gauge argument" is in fact just a convenient label for what is an amalgam
of the most common features figuring in many similar such arguments in the physics literature, both
popular and technical/text-book. (For an example of the former see, Mills 1989 or `t Hooft 1980, and
for the latter see Aitchison and Hey 1989.) Though there are certainly variations in the way the
argument is presented
and, most importantly, in the overall place and significance assigned to the
argument relative to other features of gauge theory
what we consider here are the most central
elements, the ones to which most presentations make appeal in one way or another.
2
See van Fraassen 1989 and Kosso 1999 for further discussion of the types and place of
symmetry arguments in physics.
3
Here, I suppress all spinor indices, and
is just the Dirac conjugate of and
are the usual
Dirac matrices.
4
The action will be invariant, and thus the Euler-Lagrange equations determined from minimizing
(extremizing) it covariant, if the Lagrangian is quasi-invariant, i.e., invariant up to an overall
divergence, under the transformations.
5
The homogeneous field equations follow from the local (gauge) invariance of the action, in fact
being just the identities (generalized Bianchi identities) following from Noether's 2nd theorem.
6
As was necessary in formulating the Standard Model, one can through a spontaneous
breakdown of the vacuum arrive at gauge field(s) with effective mass(es) without this spoiling the
gauge invariance of the underlying Lagrangian.
7
This argument generalizes to fields carrying symmetries associated with arbitrary (in particular,
non-Abelian) Lie groups, this yielding further interesting gauge structure. The chief difference is that
the non-Abelian gauge group has the result that the gauge field "generated" in the gauge argument
carries its own charge and is thus self-interacting. Specifically, the kinetic term necessarily includes
self interactions in the form of a term proportional to the commutator of the gauge potentials. The
requirement of local gauge invariance similarly determines the form of the (self-) interactions in the
non-Abelian case.
8
The gauge argument begins from a globally gauge invariant Lagrangian
in general, one
specifically chosen to describe a field with a known converved current, which in turn is related via
Noether's 1st theorem to invariance of the action under some compact simple Lie group.
9
Auyang 1995 contains a more developed argument along what I believe are similar lines.
Auyang's argument against global symmetries, and for local symmetries, hinges on the assumption
that a global symmetry of some field system implies the appeal to a global convention which is in
turn associated with a single internal state space for the entire field. I think this assumption can
safely be rejected.
10
See Wald and Lee 1990 for further discussion.
11
See, for example, Ramond 1990. In quantum field theory, on the other hand, locality is
commonly formalized in terms of micro-causality, the commuting of space-like separate local
algebras of observables. Clearly, there are numerous notions of "locality," some appropriate to only
certain formal contexts.
12
See, for example, Ryder 1996, 93.
13
One might also consider the apparently weaker requirement of the Lorentz invariance of the
action and corresponding Lorentz covariance of the associated equations of motion. But, the
requirement of Lorentz invariance does not exclude Lagrangians with global invariances: e.g., the
Lagrangian for the free Dirac field.
14
This, of course, leaves open the possibility that a viable locality argument of the above sort
could be mounted if the gauge transformations were afforded some non-trivial physical significance.
15
According to a prominent modern view this is not quite correct. I discuss below further matters
relevant to the exclusion of such terms.
16
See Norton 2000 for some critical discussion of considerations of simplicity.
17
Strictly speaking, this is true only locally. There are potential global issues as are evidenced,
for example, in the familiar Bohm-Aharonov effect.
18
Note that the q appearing in the gauge transformations are normalization constants for the
generators of the associated Lie group and can be scaled away. This is not to be identified with the
electric charge e until one adds the kinetic term which integrally contains e.
19
I take it that it is in recognition of essentially this same point that Auyang, considering the
gauge principle in electrodynamics, remarks: "It does not stipulate an interaction field but rules
against its a priori exclusion . . ." (Auyang 1995, 58). See also Brown 1999.
20
The appeal of local gauge symmetry principles took root in a historical context in which such
heuristic guides for "generating" physical field theories were of immediate practical importance. See
Martin 2002 for further discussion here.
21
See Wald 1986; Deser 1970, 1987. Weinberg 1995, chaps. 5 and 8, in contrast with the flow of
the gauge argument, starts from a quantum theory of massless spin one fields and arrives at a
gauge invariant coupling to matter.
22
See Froggatt and Nielsen 1991, 123; Foerster, Nielsen, and Ninomiya 1980; Cornwall, Levin,
and Tiktopoulos 1974; Weinberg 1974a, 1974b and the references contained therein. See also
Weinberg 1965.
23
Formally, this result goes by the name of the decoupling theorem.
24
Froggatt and Nielsen 1991, ch. vii discusses a "random dynamics" program wherein one
desideratum is the derivation or explanation of (the usually assumed) Lorentz invariance and gauge
invariance from underlying randomness (or, effectively, a lack of assumption about what is going on)
at the fundamental level.
25
See footnote 21. Redhead 1975, for example, discusses gauge symmetries in this way as
heuristic symmetries.
26
One might consider that much of the rhetoric surrounding gauge principles is firmly entrenched
in a specifically axiomatic view of physical theory. And one might take it that the rhetoric here also
commits the sin of taking the language of such an axiomatization too seriously, reifying features of it
as features of the world. Perhaps a semantic view of theories that, moreover, resists such a
reification would go some way toward providing a basis for a deflation of the rhetoric here.
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