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Gauge symmetry: its virtues and vices don't contradict each other

Three physicists affiliated with Princeton (now or recently) published an interesting preprint,

Locality and Unitarity from Singularities and Gauge Invariance
I know Nima from Harvard very well, he's brilliant and fun. Jaroslav Trnka is a big mind and my countrymate. Although I am a French writer (a month ago, I had to memorize sentences like "Je suis un écrivain français" for my sister's BF, one of the 21 cops who shot the terrorist in Nice), I only know that both Laurentia and Rodinia were supercontinents about 1 billion years ago.

Laurentiu Rodina is a particularly interesting hybrid name of an author especially because the supercontinent Laurentia (basically Eastern 2/3 of North America now) was a portion of the supercontinent Rodinia. Laurentia was named after the St Lawrence River which was named after Lawrence of Rome. Rodinia is named after Rodina – a Slavic word meaning "the motherland" in Russian but "the family" in Czech. Yes, this "subtle difference" appears on the Czech-Russian edition of the false friends of a Slavist.

At any rate, the Rodinia was a motherland or a family of smaller supercontinents that included Laurentia. (Rodinia was a more ancient counterpart of Pangaea – a clumping of all continents into one – except that Pangaea existed between 300 and 200 million years ago, much more recently.) There's some redundancy in Laurentiu Rodina's name – and this redundancy and the subtleties linked to it may be similar to those of the gauge symmetry.

OK, after this silly geological introduction, we are finally getting to theoretical physics.




A nasty crackpot who was very influential 10 years ago recently claimed that by co-authoring the new paper, Nima Arkani-Hamed has made a big U-turn because he liked to dismiss the gauge symmetry.

Well, this criticism is inadequate for two reasons. First, as the current Czech president likes to say, only a moron never changes his opinions. Indeed, physicists are often fortunate to make great advances that prove their old opinions incorrect and open the path towards a much deeper or more accurate truth. Good scientists do care about the evidence and proving that they have been morons is actually one of their favorite sports, at least for some of them.

Second, no change of the opinions was needed in this case because the "old and seemingly dismissive" comments about the concept of gauge symmetry don't really contradict the "new and flattering" adjectives. Gauge symmetry is really a redundancy, not a physical symmetry – and it is very useful and may have deep implications for the existence of the spacetime and the form of laws of physics, too.




Needless to say, the Standard Model of particle physics – the last undisputed "theory of nearly everything" describing all non-gravitational phenomena – is tightly linked to the concept of the gauge symmetry. At every point \((x,y,z,t)\) of the spacetime, we may pick an element \(g(x,y,z,t)\) of the gauge group which is basically \(SU(3)\times SU(2)\times U(1)\), and perform a transformation of the fields. The fields generally change. The fields transform in various representations of the gauge group (products of singlets, doublets, or triplets under both \(SU(2)\) and \(SU(3)\), marked by particular values of the hypercharge \(Y\) generating the \(U(1)\) – all irreps of \(U(1)\) are one-dimensional). On top of the expected transformation rules, the gauge fields \(A_\mu(x,y,z,t)\) also get modified by terms proportional to \(g^{-1}(x,y,z,t) \partial_\mu g(x,y,z,t)\) – which take values in the Lie algebra of the gauge group, just like the gauge fields.

The gauge symmetry is the requirement that when the Standard Model fields obeyed the (field) equations of motion before the transformation, the transformed fields obey them, too. The previous sentence sounds like a standard physicist's definition of a symmetry. If something obeys the laws of physics before the transformation, it does obey them after the transformation, too.

There is a reason why we say that the gauge symmetry is not a real symmetry. The "something" before the transformation and the "something" after the transformation are actually objects that must be physically identified with each other. When you make any measurement by your apparatuses, they will produce the same results whether or not the transformation took place. The untransformed configuration of the fields and the transformed one only differ in "physically unmeasurable" quantities.

To be specific, in electromagnetism, the electric and magnetic vectors \(\vec E(x,y,z,t)\) and \(\vec B(x,y,z,t)\) may be measured by some gadgets. And they also happen to be gauge-invariant – their numerical values don't depend on the gauge transformation parameter \(g(x,y,z,t)\) I mentioned above. They're the same "before" and "after". On the other hand, the gauge potentials \(\phi(x,y,z,t)\) and \(\vec A(x,y,z,t)\) do change after the transformation. But they cannot be measured. The value of the potential depends on your "gauge choice", "calibration", i.e. basically on human conventions.

This situation differs from global symmetries. When you rotate an object (a spaceship with a laboratory) by the angle \(\alpha\), it may obey the laws of physics after the rotation if it obeyed them before it. The astronauts inside the spaceship may be unable to determine whether their spaceship is a rotated one or not – after all, the word "rotated" should be supplemented by "with respect to whom", they may object. However, it makes sense to distinguish the rotated and unrotated spaceship. An external observer really has to distinguish them because his relative orientation with respect to the spaceship may be measured. That's why the rotation of a spaceship is a true, "global" symmetry.

On the other hand, the relative orientations are equally unmeasurable in the case of the gauge symmetry. You may choose the value of \(g(x,y,z,t)\) in the gauge group that is nontrivial inside a spaceship but trivial outside the spaceship. But the observer outside the spaceship will still be unable to distinguish the untransformed and transformed state. They're physically identical.

In a quantum mechanical theory, if the states \(\ket\psi\) and \(\ket\psi + \epsilon G \ket\psi\) of quantum fields are physically identical, it means – by linearity – that the state \(\epsilon G\ket\psi\) is physically identical to the zero vector of the Hilbert space. Omit the coefficient: \(G\ket\psi\) must be physically identical to zero. Such "pure gauge" states (the variations of some states under infinitesimal gauge transformations) must behave as unphysical ghosts of a sort. Physics allows ghosts and angels on needles as long as they behave: They are not allowed to mess with the experiments (and with Texas). The condition I mentioned is equivalent to the requirement that the projection of \(G\ket\psi\) into the physical Hilbert space is zero. If \(G\ket\psi\) were not zero, it would mean that the state carries some "charges" (or "charge densities") under the gauge group i.e. that the state is non-invariant, a non-singlet. A physical state must be a singlet (invariant) under the gauge symmetry, however! That's different from a true, global symmetry: Physical states are allowed to be (and usually are) non-singlets i.e. non-invariant under the global symmetries. For example, planets carry a nonzero angular momentum even though \(\vec J\) generates a (global) rotational symmetry.

One may show that it means that these states are null states (their norm is zero, as expected e.g. from null polarizastions of a photon, in this case \(\epsilon^\mu\sim k^\mu\)). But they must also be decoupled from the physical states we care about. If you guarantee that states not orthogonal to such pure states are absent in the initial state, those must be absent in the final state, too. Gauge symmetry removes two polarizations out of four that could existed for a photon field \(A_\mu(x,y,z,t)\). One of them has \(\epsilon^\mu\sim k^\mu\) and is the "pure gauge" state that is null and harmless. The other one is a state obeying \(k^\mu \epsilon_\mu \neq 0\); which of those vectors is chosen is irrelevant. This other unphysical state must be banned in the initial state by Gauss' law (a non-dynamical equation of motion such as the Maxwell's equation \({\rm div}\,\vec D = \rho\) – "non-dynamical" means "not containing time derivatives") and the gauge symmetry (more or less equivalent to a conservation of the charge, thanks to Emmy Noether's theorem) basically guarantees that if there's no violation of Gauss' law in the initial state, there won't be any violation in the final state.

So you may consistently demand that the "evil unphysical" polarizations of the photon, those with \(\epsilon_\mu k^\mu \neq 0\) that could be sensitive to the "harmless unphysical" polarizations \(\epsilon^\mu \sim k^\mu\), are never produced by the scattering or other physical processes (in evolution). That's needed for your ability to consistently demand that they're absent in all (initial and final) states.

(In the BRST formalism, the BRST-exact states \(Q\ket\lambda\) explain why the polarizations created with \(\epsilon^\mu\sim k^\mu\) as well as those created by the \(c\)-ghost are unphysical, while the BRST-non-closed states \(Q\ket\psi \neq 0\) may be consistently forbidden, which is what eliminates the unphysical \(k^\mu \epsilon_\mu \neq 0\) states as well as those created with the \(b\)-antighost. The BRST formalism makes many loop calculations elegant if the symmetry is non-Abelian – but the whole BRST formalism and the new \(b,c\) fields added to it become pretty much worthless for an Abelian symmetry.)

Gauge symmetry kills negative-norm states

OK, I have implicitly explained why the gauge symmetry is "needed". It kills unphysical polarizations of the photon and quanta of other spin-1 (or higher) fields – and those could be harmful. The polarization with \(\epsilon^\mu \sim k^\mu\) is null (probabilities are equal to zero by Born's rule) and harmless by itself. But the polarizations with \(\epsilon^\mu k_\mu \neq 0\) are "sensitive" (not orthogonal) to the harmless null polarizations and that could be dangerous. These dangerous polarizations would behave as psychics who can feel the angels on the needle – and it's really the psychics, not the angels, who are dangerous because the angels and psychics change the probabilities by zero and nonzero, respectively. ;-)

In effect, that would force us to keep the time-like polarization with \(\epsilon^\mu = (1,0,0,0)\) in the spectrum, because it's one of the harmful non-orthogonal polarizations that cannot be consistently removed, and because gauge bosons with this time-like polarization possess a negative-norm, some processes that include them would be predicted to occur with negative probabilities. That would be trouble for the LHC experimenters because most of them are unable to observe minus one million collisions of a certain kind. ;-)

The gauge symmetry removes the "harmless" null longitudinal polarization, \(\epsilon^\mu\sim k^\mu\), as well as the "harmful" polarizations such as the time-like one. That's great because only the polarizations in the \(x\)- and \(y\)-directions – with \(\vec k\) in the \(z\)-direction – are kept in the physical spectrum.

So if we need a consistent theory which predicts non-negative probabilities and we want to use a quantum field \(A_\mu(x,y,z,t)\) with a Lorentz index (i.e. fields of spin equal to one or greater than one), we simply need a gauge symmetry with the same number of generators to cure the potential diseases. In other words, the gauge symmetry is needed in every manifestly Lorentz-invariant quantum field theory with elementary particles of spin \(j\geq 1\).

The need for the gauge symmetry is fatally important if the conditions are met. In particular, a gauge anomaly (a one-loop quantum process that violates a symmetry that used to hold at the tree level) means an inconsistency of a theory with a gauge symmetry because such an anomaly would prevent us from consistently banning the psychics. (Only when I was editing the blog post, I realized that the terminology involving angels-and-psychics for the two unphysical polarizations should be used much more systematically, even in textbooks and courses.)

Note that the gauge symmetry is only "needed" if both conditions are satisfied. There must be some spin-1 or greater particles. And we want a formalism where the Lorentz symmetry is manifest. If you violate either condition, the need for the gauge symmetry goes away. If you have a quantum field theory with scalar and spinor fields only, \(j=0\) and \(j=1/2\), you won't need a gauge symmetry. And even if you include spin-1 particles, you may get rid of all the gauge fields by working with some gauge-fixing condition and eliminating the unphysical polarizations from scratch.

There exist more or less elegant methods to achieve it – to get rid of the unphysical polarizations of photons created by \(A_0\) and \(A_z\), if you wish, and the corresponding gauge symmetry that makes them harmless. In the twistor-based business, the gauge symmetry was basically non-existent from scratch because only the (two transverse) physical polarizations naturally arise in the twistor formalism. So while the removal of the unphysical polarizations and the corresponding gauge symmetry may look messy in the normal Lorentz-index-based formalisms, it's the other way around in the twistor formalism. Only the two transverse, \(x,y\) physical polarizations are natural in the twistor formalism. The remaining two are unnatural and unnecessary.

Arkani-Hamed et al. have learned lessons from their extensive work on the twistor formalism – where the gauge symmetry doesn't arise at all which supports the dismissive claims about the gauge symmetry. But they mentally returned to the normal non-twistor spacetime with Lorentz indices that most of us are familiar with and imported some wisdom from the twistor world. And they claim that some amplitudes may be fully reconstructed from some singularities and the gauge symmetry. This is basically analogous to a claim in the twistor world that all amplitudes are obtained from some singular ones by recursive formulae. In the non-twistor world, there are new degrees of freedom – the unphysical polarizations of the fields – that could destroy the uniqueness of the answers. But if one adds the corresponding requirement of the gauge symmetry, this non-uniqueness goes away. That's pretty much why the paper they released has to work.

Most of us who have ever thought about standard particle physics consider gauge symmetry to be a "master" principle. Even though it is not a real, global symmetry, its choice determines a large portion of the physical content of a quantum field theory. Well, the choice of the gauge symmetry in the gauge-symmetric formalism basically determines the list of elementary spin-1 particles (and if we generalize the Yang-Mills symmetry to local supersymmetry in supergravity or local diffeomorphisms in general relativity, we may also talk about spin-3/2 and spin-2 particles) and their tree-level cubic and quartic interactions. Moreover, the gauge symmetry invites us to classify the remaining fields as representations of the gauge group which organizes the other fields and determines their interactions with the gauge fields, too. It's pretty.

It's pretty but it's not "absolutely needed". A formulation without an explicit Lorentz invariance may avoid the gauge symmetry. Moreover, such a formulation may be needed because the gauge invariance is only needed for a Lorentz-covariant treatment of elementary spin-1 particles and one might conjecture that all the spin-1 particles may be composite in some sense and there's no truly universal litmus test to distinguish elementary and composite particles. (Well, there are some theorems banning composite massless or other spin-1 bosons under certain assumptions but there surely exist spin-1 composite bosons in QCD, among other things, and those could emulate at least massive gauge bosons under certain circumstances.)

While it's indisputable that the gauge symmetry is useful as an organizing principle to build important theories; and that it's not quite necessary in physics, a pragmatic question remains: Will you miss anything if you choose one of the extreme viewpoints? You may view gauge symmetries as accidents that have helped us to find some theories but those can be formulated without gauge symmetries as well. In other words, you may downplay the importance of the gauge symmetries in physics.

Alternatively, you may say that gauge symmetries really are important and will keep their place in the future of physics. You may dismiss all the "gauge symmetry is just a redundancy" talk as some irrelevant babbling that only led to the awkward definition of theories that should "naturally" be defined with the gauge symmetries, anyway. After all, you may argue (and I often do argue) that a subset of gauge symmetries (which change the fields at infinity, typically by a "constant" transformation) does behave as a set of true global symmetries and non-singlet/charged states under these transformations should be considered physical – so at least if you use fields (degrees of freedom) that allow you to talk about both kinds of symmetries, you can't really throw away all the gauge symmetries without throwing away the truly physical global symmetries.

I am not sure about the answer – and I think that no one has a proof that settles the answer – and I guess that Nima and others are sort of open-minded, too. Depending on the available evidence and fresh new arguments, calculations, and ideas, one's opinions may drift from "gauge symmetry will survive in the future of physics" to "gauge symmetry will fade away" and back. In particular, Nima likes to change his perspective in rather extreme ways – it's perhaps a part of the personality. He may have made a U-turn concerning the anthropic principle (perhaps a few times) and he's happy about the freedom that science allows to passionate yet flexible physicists like himself. ;-) Just to be sure, I do think that true science allows this attitude – but it doesn't really require it. There exist perspectives from which Nima is a conservative physicist – every great physicist has to be conservative according to a suitable definition of conservativeness. On the other hand, there are perspectives from which Nima is a classic revolutionary if not "chaos maker". Both attitudes may be useful when one is smart and lucky, of course.

There are various indications that the gauge symmetry could be rather fundamental. For example, in a January 2015 paper, Mintun, Polchinski, and Rosenhaus have argued that the gauge symmetry plays a truly fundamental role in the "quantum error correction" mechanism that allows gravity to be holographically encoded. There exist other papers – whose authors mostly don't read each other – with a similar message. I really do think that they should talk to each other much more than they do.

I am actually often colliding with this topic of "gauge symmetry as a master principle of the spacetime" in the part of my research focusing on the emergence of the spacetime. One reason why this topic is omnipresent in my approach is that the pure gauge polarizations of gauge bosons ultimately arise from the Virasoro-exact states of strings in perturbative string theory, \(L_{-1}\ket\psi\), roughly speaking, so all spacetime gauge symmetries are "offspring" of the key gauge symmetry on the string world sheet, the conformal symmetry. I often tend to imagine that a "more abstract and complex" symmetry like the conformal one exists even nonperturbatively, \(U(N)\) in the BFSS matrix model is a close cousin, and the identification of the right symmetry basically "implies" the right identification of all the spacetime gauge symmetries as well as the difference between physical and unphysical polarizations, and therefore the causal structure of the spacetime and other things, too.

And the gauge symmetry is also important because Wilson lines may be nonlocal, and sometimes heavily nonlocal, degrees of freedom that determine the geometric relationships between regions that may be a priori connected or disconnected etc. To say the least, I believe that the gauge theories with a manifest gauge-covariant formalism encourage (and maybe are needed) for the definition of truly natural non-local degrees of freedom in local theories – and those may be helpful to discuss wormholes and entanglement in quantum gravity and related issues.

So I think that there's a lot of potential for the gauge symmetries to become less important than they seem today – and lots of potential for them to be more important than they seem today. I am not sure about the future evolution of their status in the world class physics research. But what I am sure about is that Nima's statements weren't ever meant to say that 0% or 100% of the future papers in theoretical physics will allow an important role to be played by gauge symmetries. The percentage is unknown and almost certainly strictly in between 0% and 100%. The statement that "the gauge symmetry isn't a real symmetry" was always meant to say something very specific and something that has been settled by a proof, one that shold be understandable to a good student.

People who don't understand any of these "nuances" should better use their opportunity to shut up. Theoretical physics is surely not a kindergarten pissing contest where one side mindlessly shouts "gauge symmetry akbar" and the other side shouts critical slogans about the gauge symmetry. To follow theoretical physics of 2016, you simply need to learn much more than the four words "gauge symmetry, yes, no".

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