Internal (gauge) symmetries and spacetime symmetries

[TrickyDicky]

Internal symmetries of the SM -U(1), SU(2), SU(3)- are usually said to belong to abstract spaces unrelated to spacetime symmetries, have there been many attempts to relate internal symmetries to spacetime symmetries, and if so how far have they gotten?

 

[Bill_K]

Are you thinking of the Coleman-Mandula Theorem?

 

[TrickyDicky]

Are you thinking of the Coleman-Mandula Theorem?

Hmm, I had never heard of it, but i just looked it up on wikipedia. After reading its description it seems to have so many limitations that it is hardly a theorem that anyone could take seriously.
I was thinking about really trying to come up with ways to bring closer the SM abtract internal symmetries to the actual spacetime symmetries.
Say for instance U(1) phase invariance can be integrated within the broader spherical symmetry, or maybe SU(2) being topologically a hypersphere could somehow be related to spherical symmetry too; I'm just making up the examples so someone might get the drift of what I'm referring to. I know the particular U(1) and SU(2) of the SM are not exactly the global ones but local gauges related to weak hypercharge and weak isospin.

 

[tom.stoer]

The Coleman-Mandula theorm says that spacetime symmetries and internal symmetries cannot be combined into one larger symmetry structure (the only possibility is the direct product which so to speak trivial). But there is one loophole, namely 'fermionic charges'. So using SUSY one can indeed combine spacetime symmetries and internal symmetries. And when gauging SUSY (i.e. when it becomes a local symmetry) you get super-gravitation SUGRA

 

[TrickyDicky]

The Coleman-Mandula theorm says that spacetime symmetries and internal symmetries cannot be combined into one larger symmetry structure (the only possibility is the direct product which so to speak trivial). But there is one loophole, namely 'fermionic charges'. So using SUSY one can indeed combine spacetime symmetries and internal symmetries. And when gauging SUSY (i.e. when it becomes a local symmetry) you get super-gravitation SUGRA

Yes, I was aware of SUSY, SUGRA and the Kaluza-Klein theories, and indeed all these can be classified as serious attempts to do what I asked in the OP. But I see them as theories more concentrated on unification of already known theories and interactions into a larger framework like a TOE. Besides, so far they all seem to lack any empirical support.
I was rather thinking about a more purely geometrical approach, we know by the outcome of the high energy experiments that the SM internal symmetries are there, they have to somehow be related to the known properties of our spacetime or at least be manifestations of the spacetime symmetries in the sense that there might be an underlying pattern that unifies the internal symmetries and that we are interpreting them in a fragmented way.
I'm afraid that the success of the SM may end up locking it to any further possibility to be combined with gravitation and the observed spacetime symmetries.

 

[tom.stoer]

I was rather thinking about a more purely geometrical approach, we know by the outcome of the high energy experiments that the SM internal symmetries are there, they have to somehow be related to the known properties of our spacetime or at least be manifestations of the spacetime symmetries in the sense that there might be an underlying pattern that unifies the internal symmetries and that we are interpreting them in a fragmented way.

I know about the following approaches pointing towards something like thast, namely
- string theory (which would mean unification)
- SUGRA (which would mean unification, too)
- non-commutative geometry (?)

 

[TrickyDicky]

I know about the following approaches pointing towards something like thast, namely
- string theory (which would mean unification)
- SUGRA (which would mean unification, too)
- non-commutative geometry (?)

Ok, I'll look up the non-commutative geoemtry approach, there is something in wikipedia. If I have questions I might take them to the BSM subforum.
Thanx

 

[strangerep]

[...] we know by the outcome of the high energy experiments that the SM internal symmetries are there, they have to somehow be related to the known properties of our spacetime or at least be manifestations of the spacetime symmetries in the sense that there might be an underlying pattern that unifies the internal symmetries [...]

No, no, and no. The crucial point here is that gauge transformations are unphysical. Experiments verify predictions of the SM, i.e., that the SM Lagrangian is right in a vast number of ways. But the gauge freedoms are unphysical extra degrees of freedom in the Lagrangian. All observable results are gauge-independent.

In contrast, spacetime transformations of the Poincare group are physical -- we can physically perform rotations, translations, etc.

Once this distinction is understood, the Coleman-Mandula theorem becomes more comforting, and less of a nuisance.

 

[tom.stoer]

There are of course formulations of GR (or its extensions like Einstein-Cartan) where the Lorentz- or the Poincare group are gauged, i.e. where Lorentz symmetry becomes a local gauge symmetry (most famous: Ashtekar variables in LQG). In that sense the language in which GR and gauge theories like the SM are formulated are rather similar. GR is so to speak a gauge theory with a local Lorentz symmetry + diffeomorphism invariance and therefore with a different dynamics. But the kinematical framework is rather similar in the above mentioned formulations: local gauge symmetry, connection variables, A- and E-fields, fiber bundles, Gauss law constraint as generator of local gauge transformations, elimination of unphysical degrees of freedom (even in GR you have to project out gauge d.o.f. to reduce the 10 components of the metric to the two physical polarizations). The big difference is that in GR the gauge symmetry is intertwinded with local diffeomorphism invariance whereas in gauge theories (like the SM) formulated on flat Minkowski spacetime this aspect is trivial.

 

[PhilDSP]

No, no, and no. The crucial point here is that gauge transformations are unphysical. Experiments verify predictions of the SM, i.e., that the SM Lagrangian is right in a vast number of ways. But the gauge freedoms are unphysical extra degrees of freedom in the Lagrangian. All observable results are gauge-independent.

Doesn't that depend on the topology of the problem? What you say should be true if there are no boundary conditions, for instance, for a free solitary particle, but not in general don't you think?

Another approach to the original question is Penrose and Rindler's Twistor formulation. Rather than dealing with with Minkowski space directly, a mapping is made to Twistor space which works more consistently and more flexibly with SU(n). Ward seems to be an excellent guide to that methodology.

Has anyone seriously pursued the possibility that space-time symmetries are emergent in U(1) (as is classical electrodynamics), but that a richer set of space-time symmetries might exist with more primitive relationships in SU(n)? I suppose Kaluza-Klein theory qualifies in that regard possibly.

 

[TrickyDicky]

No, no, and no. The crucial point here is that gauge transformations are unphysical. Experiments verify predictions of the SM, i.e., that the SM Lagrangian is right in a vast number of ways. But the gauge freedoms are unphysical extra degrees of freedom in the Lagrangian. All observable results are gauge-independent.

In contrast, spacetime transformations of the Poincare group are physical -- we can physically perform rotations, translations, etc.

Once this distinction is understood, the Coleman-Mandula theorem becomes more comforting, and less of a nuisance.

That distinction is understood alright, but your reply makes clear you are conflating gauge transformations with gauge symmetries, gauge symmetries (gauge invariance) are key to understand gauge theories and therefore are physically relevant. In my previous posts I only discussed symmetries, not the transformations themselves.
On the other hand a spacetime symmetry is what makes that, for instance in the case of rotational symmetry, there is nothing that physically distinguishes one point along the spherical rotation from another.
In the case of internal symmetries the transformation is performed in an abstract space, and only in that sense is "unphysical", so I was precisey highlighting the distinction you mentioned to learn of possible ways of understanding those abstract symmetries in terms of physical symmetries.

 

[TrickyDicky]

Has anyone seriously pursued the possibility that space-time symmetries are emergent in U(1) (as is classical electrodynamics), but that a richer set of space-time symmetries might exist with more primitive relationships in SU(n)? I suppose Kaluza-Klein theory qualifies in that regard possibly.

U(1) is the circle group, how would spacetime symmetries other than the trivial SO(2) emerge from it?

 

[TrickyDicky]

There are of course formulations of GR (or its extensions like Einstein-Cartan) where the Lorentz- or the Poincare group are gauged, i.e. where Lorentz symmetry becomes a local gauge symmetry (most famous: Ashtekar variables in LQG). In that sense the language in which GR and gauge theories like the SM are formulated are rather similar. GR is so to speak a gauge theory with a local Lorentz symmetry + diffeomorphism invariance and therefore with a different dynamics. But the kinematical framework is rather similar in the above mentioned formulations: local gauge symmetry, connection variables, A- and E-fields, fiber bundles, Gauss law constraint as generator of local gauge transformations, elimination of unphysical degrees of freedom (even in GR you have to project out gauge d.o.f. to reduce the 10 components of the metric to the two physical polarizations). The big difference is that in GR the gauge symmetry is intertwinded with local diffeomorphism invariance whereas in gauge theories (like the SM) formulated on flat Minkowski spacetime this aspect is trivial.

I was thinking about this parallelism when writing the OP.
However many people don't accept this analogy, and don't consider GR as a gauge theory. Most of them mention the distinction between passive and active coordinate transformations in dynamic theories like GR.

 

[TrickyDicky]

After reading the wikipedia article on noncommutative SM (suggested by Tom stoer), it turns out that there is a guy named Alain Connes , a french mathematician that has been trying to do exactly what I had in mind, with not much success by the way. A brief search using his name shows his work has been frequently discussed in PF.
See http://www.scientificamerican.com/article.cfm?id=the-geometer-of-particle and
http://resonaances.blogspot.com.es/2007/02/alain-connes-standard-model.html

 

[tom.stoer]

In the case of internal symmetries the transformation is performed in an abstract space, and only in that sense is "unphysical", ...

That's not what we mean.

The flavor symmetry is defined in an abstract space, it's an internal symmetry, but it's not unphysical!

The color-gauge symmetry SU(3) or the U(1) symmetry in electrodynamics are internal symmetries, too, but they are indeed unphysical.

The reason is that they can be gauge-fixed which goes hand in hand with reduction of unphysical do.f. Gauge fixing in QED is well known (it does not mean to break the invariance!) and it removes the unphysical d.o.f., i.e. it eliminates 2 unphysical photons, i.e. 2 physical transversal polarizations remain.

That is the reason why gauge symmetries are unphysical; they operate on unphysical d.o.f. (of the gauge fields).

 

[strangerep]

No, no, and no. The crucial point here is that gauge transformations are unphysical. Experiments verify predictions of the SM, i.e., that the SM Lagrangian is right in a vast number of ways. But the gauge freedoms are unphysical extra degrees of freedom in the Lagrangian. All observable results are gauge-independent.

Doesn't that depend on the topology of the problem? What you say should be true if there are no boundary conditions, for instance, for a free solitary particle, but not in general don't you think?

Afaik, one typically assumes that the gauge transformations are trivial at (eg) spatial infinity, and at the temporal endpoints of the action integral (the rationale being that the field configurations at those endpoints are physical, hence must remain unaffected by gauge transformations).

Another approach to the original question is Penrose and Rindler's Twistor formulation. Rather than dealing with with Minkowski space directly, a mapping is made to Twistor space which works more consistently and more flexibly with SU(n).

The trouble I've always had with the Twistor formulation is that you're basically dealing with the representation theory of the conformal group, or rather, it's quad-cover SU(2,2). But the conformal Casimirs are very different from Poincare, and I've yet to see an attractive way to make them "play nice" together. But this must somehow be done in order to construct suitable asymptotic states and only then confront Coleman-Mandula.

 

[strangerep]

[...] but your reply makes clear you are conflating gauge transformations with gauge symmetries, [...]

?

I'd always understood the term "gauge symmetry" to be associated with a group of "gauge transformations", and "gauge invariant" to be an adjective describing quantities which remain invariant under those transformations.

Apparently, you have some other definition of "gauge symmetry" in mind, but I have no idea what that might be.

 

[TrickyDicky]

That's not what we mean.

The flavor symmetry is defined in an abstract space, it's an internal symmetry, but it's not unphysical!

The color-gauge symmetry SU(3) or the U(1) symmetry in electrodynamics are internal symmetries, too, but they are indeed unphysical.

The reason is that they can be gauge-fixed which goes hand in hand with reduction of unphysical do.f. Gauge fixing in QED is well known (it does not mean to break the invariance!) and it removes the unphysical d.o.f., i.e. it eliminates 2 unphysical photons, i.e. 2 physical transversal polarizations remain.

That is the reason why gauge symmetries are unphysical; they operate on unphysical d.o.f. (of the gauge fields).

I think we're talking at cross-purposes here. Please, read again my post, I made a clear distinction between the symmetry groups and the individual transformations. The physical versus unphysical distinction in those posts was referred to the transformations, not to the symmetries.
I do understand that when you talk about unphysicality of gauge symmetries you are merely referring to the reduction of d.o.f one can perform by fixing gauges; for instance in the analogy you make with gauge symmetry in GR, we say coordinates are unphysical in that sense because general covariance makes sure the group of isometries acts as a gauge symmetry. However you wouldn't call the active isometry unphysical since it can be performed in spacetime like strangerep commented when he mentioned translations in spacetime as something physical.
Now in the case the SM gauge groups U(1)XSU(2)XSU(3) their symmetries apparently belong to some abstract space unrelated to our spacetime. Your example of flavor symmetry is not a gauge symmetry, it is the SU(3) flavor symmetry that is related to (has as subgroups) SU(2) (isospin) and U(1) (hypercharge), these are different from the above mentioned:weak isospin SU(2), weak hypercharge U(1) (that most current references refer to just as isospin and hypercharge which could lead to confusion if the cotext is not well fixed) and color charge SU(3). My discussion was restricted as can be seen in the title of the OP to the internal gauge symmetries, not internal symmetries in general.

 

[TrickyDicky]

?

I'd always understood the term "gauge symmetry" to be associated with a group of "gauge transformations", and "gauge invariant" to be an adjective describing quantities which remain invariant under those transformations.

Don't worry we have the same concept of gauge symmetry.

My point there was that you were (probably inadvertently) not making the distiction between them to reply my post since you quoted me talking about the gauge groups, not the individual transformations that make up the group; they are certainly intimately associated, but, a gauge transformation is not a gauge symmetry group. It is part of it. This really goes nowhere since we have already cleared up in posterior posts what everyone means by gauge symmetry and by "unphysical". And I think we all agree.

 

[PhilDSP]

The trouble I've always had with the Twistor formulation is that you're basically dealing with the representation theory of the conformal group, or rather, it's quad-cover SU(2,2). But the conformal Casimirs are very different from Poincare, and I've yet to see an attractive way to make them "play nice" together. But this must somehow be done in order to construct suitable asymptotic states and only then confront Coleman-Mandula.

Interesting, I haven't looked at it in enough detail yet to see those types of limitations. Twistor theory doesn't seem to have retained the momentum it had 10 or 20 years ago in the literature, so it may have encountered impasses too difficult to sort out at the present time.

 

[PhilDSP]

U(1) is the circle group, how would spacetime symmetries other than the trivial SO(2) emerge from it?

It's not so much that U(1) would produce the symmetries but rather that U(1) would be a reduction of a richer set (from SU(n) or some other algebra). In situations where the richer symmetries are broken you get a more simple set, though not necessarily a subset. In Kaluza-Klein theory, for example, mass is an additional dimension that is factored along with space and time to get a more complex set of symmetries than space-time symmetries. If you regard c as a constant in the equation $E = mc^2$ then neither time nor space enter as dimensions on the right hand side of the equation. That should indicate the presence of other symmetry relationships somehow underlying the equation.

 

[TrickyDicky]

It's not so much that U(1) would produce the symmetries but rather that U(1) would be a reduction of a richer set (from SU(n) or some other algebra). In situations where the richer symmetries are broken you get a more simple set, though not necessarily a subset. In Kaluza-Klein theory, for example, mass is an additional dimension that is factored along with space and time to get a more complex set of symmetries than space-time symmetries. If you regard c as a constant in the equation $E = mc^2$ then neither time nor space enter as dimensions of the equation. That should indicate the presence of other symmetry relationships somehow underlying the equation.

Ah, ok, yes that was precisely the point of the OP, to find the possible underlying symmetries. But unlike the Kaluza case within our spacetime dimensions.

 

[strangerep]

[...]I had never heard of [Coleman-Mandula thm], but i just looked it up on wikipedia. After reading its description it seems to have so many limitations that it is hardly a theorem that anyone could take seriously.

Why do you say that? I always thought the inputs to the CM thm were reasonable.

 

[strangerep]

My point there was that you were (probably inadvertently) not making the distinction between them to reply my post since you quoted me talking about the gauge groups, not the individual transformations that make up the group; they are certainly intimately associated, but, a gauge transformation is not a gauge symmetry group. It is part of it. This really goes nowhere since we have already cleared up in posterior posts what everyone means by gauge symmetry and by "unphysical". And I think we all agree.

Well, then I still don't understand the point of the distinction you're trying to make (even after re-reading your earlier posts). A group is simply a set of elements, with certain multiplication rules defined thereon, sure. Perhaps you're trying to distinguish between an abstract group and concrete representations thereof?

 

[tom.stoer]

Your example of flavor symmetry is not a gauge symmetry, it is the SU(3) flavor symmetry ... My discussion was restricted as can be seen in the title of the OP to the internal gauge symmetries, not internal symmetries in general.

I know. I wanted to make clear that your concept "In the case of internal symmetries the transformation is performed in an abstract space, and only in that sense is "unphysical", ... " of symmetries which are unphysical b/c they act in some abstract space is wrong. Flavor symmetry acts on an abstract space and is physical; gauge symmetries act on abstract spaces as well, and they are unphysical. The fact that there is an abstract space has nothing to do with the fact that they are unphysical (b/c there are physical symmetries like flavor acting on abstract spaces, too).

I don't really understand what you intention is.

 

[strangerep]

Flavor symmetry acts on an abstract space and is physical;

Hmmm. Now I don't understand what you mean. In what sense is flavor symmetry physical? If you're talking about rotations in isospin space, I was under the impression that (eg) the nucleons are considered to be physically the same if the other quantum numbers like mass, electric charge, etc, are ignored. (?)

 

[martinbn]

What exactly is abstract space and (I guess) non-abstract? Rotations act on space and they act on the wave functions as well. Does that make them both physical and unphysical?

 

[TrickyDicky]

Why do you say that? I always thought the inputs to the CM thm were reasonable.

I simply meant there are many loopholes in the theorem,at least according to wikipedia, so that it can easily be avoided.

 

[TrickyDicky]

Well, then I still don't understand the point of the distinction you're trying to make (even after re-reading your earlier posts). A group is simply a set of elements, with certain multiplication rules defined thereon, sure. Perhaps you're trying to distinguish between an abstract group and concrete representations thereof?

Forget it, I just wanted to make you see that when you
replied:"No, no and no" to my post #5 you were missing my point, that had nothing to do with the physicality of gauge symmetries.
My point there was just about doing what Alain Connes is trying to with his noncommutative SM. See the links I posted.

 

[TrickyDicky]

I know. I wanted to make clear that your concept "In the case of internal symmetries the transformation is performed in an abstract space, and only in that sense is "unphysical", ... " of symmetries which are unphysical b/c they act in some abstract space is wrong. Flavor symmetry acts on an abstract space and is physical; gauge symmetries act on abstract spaces as well, and they are unphysical. The fact that there is an abstract space has nothing to do with the fact that they are unphysical (b/c there are physical symmetries like flavor acting on abstract spaces, too).

I don't really understand what you intention is.

See the last part of my last
Post to understand it.
Wrt the physicality I already said it was a purely semantic thing.
I was just using the word with a different meaning than you.
Strangerep also seems not to coincide with you on that.

 

[TrickyDicky]

What exactly is abstract space and (I guess) non-abstract? Rotations act on space and they act on the wave functions as well. Does that make them both physical and unphysical?

Good point. Physical is an ambiguous term.

 

[tom.stoer]

Physical means that if you do a isospin rotation U_iso on a physical system you get a different system: U_iso|proton> = |neutron>; it's the same as if you act with a rotation R on the solar system, i.e. if you rotate the whole solar system in space; you get a different, physically allowed system R|solar system> = |rotated solar system>.

Quantum mechanically the operators U and R are unitary operators, i.e.

U[θ] = exp(iθ^aI^a)

R[α] = exp(iα^aL^a)

where I and L are the isospin and orbital angular momentum operatorsa and θ, α are the rotation angles.

Unphysical means that if you construct something like U for gauge transformations you can still do a gauge fixing; that means that the Hilbert space becomes something like

H = H_phys + H_unphys

and that for all states in H_phys the unitary operator implementing a gauge transformation g is reduced to the identity:

U[g] |phys> = id |phys>

U[g] |ψ> = |ψ>

This is different from the above described isospin case b/c for isospin (with isospin being an exact symmetry) all measurable quantities are identical, but the states are not! Mathematically there is still an iso-doublet, an iso-triplet etc. But for gauge transformations there is only a gauge-singulet. All states not belonging to the physical (= singulet sector) are unphysical, they can't be created, they do not result from time evolution, they cannot be measured or observed, they do not interact with physical states etc. So mathematically gauge fixing is somethjing like projecting to a huge physical Hilbert space; nothing like that happens for rotations, isospin etc.

Look at rotations: all theories we know are invariant w.r.t (global) rotations; but the states we observe are not! We observe different orientations in space, we can distinguish between different p-orbitals in the hydrogen atom, etc. We have observables in our theories to make this distinction. Nothing like that is possible with gauge transformations; there is no observable which allows us to distinguish between different gauge sectors!

Let's look at the generator of gauge transformations G^a. These generators commute with all observables; better: an operator A is an observable only if it commutes with these G^a.

[A, G^a] = 0

Now look at rotations and isospin; of course you can have an observable L_x which does not commute with L_y.

Regarding abstract space; I think what really matters is that this space is not ordinary 3-space!

Again let's look at ordinary rotations. In field theory you can rotate a position-space vector r using a rotation matrix R; now you can relate this to rotations of fields like scalar fields, vector fields, spinor fields etc. In all cases you get a second "rotation matrix" S now acting on the fields A(r); that means

r → r' = R r
A → S(R) A

I think you are familiar with this concept e.g. when constructin S(R) for the Dirac equation or when translating Lorentz transformations acting on spacetime to transformation laws acting on 4-vectors and 4-tensors.

For an abstract space (like isospin, color,. ... ) there is no such R acting on r. You can't relate isospin rotations or gauge transformations to transformations in position space; so there is no R, only a S.

I hope it's now clear that "physical vs. unphysical" and "abstract" have nothing to do whith each other!

 

[TrickyDicky]

Physical means that if you do a isospin rotation U_iso on a physical system you get a different system: U_iso|proton> = |neutron>; it's the same as if you act with a rotation R on the solar system, i.e. if you rotate the whole solar system in space; you get a different, physically allowed system R|solar system> = |rotated solar system>.

Look at rotations: all theories we know are invariant w.r.t (global) rotations; but the states we observe are not! We observe different orientations in space, we can distinguish between different p-orbitals in the hydrogen atom, etc. We have observables in our theories to make this distinction.

Those observables include additional info to make the distinction, and in the case of the proton and neutron rely on the fact the isospin symmetry is not exact, it is leaving aside the mass diference and the EM interaction; isn't that cheating a bit?

Regarding abstract space; I think what really matters is that this space is not ordinary 3-space!

Right.

I hope it's now clear that "physical vs. unphysical" and "abstract" have nothing to do whith each other!

I was aware that what you were calling "(un)physical" had nothing to do with what I implied in the phrase you quoted from me. Now it is even clearer.
I was using the term informally to mean "in the context of ordinary space", whuich obviously lef out all abstract spaces different from ordinary space.

 

[tom.stoer]

Those observables include additional info to make the distinction, and in the case of the proton and neutron rely on the fact the isospin symmetry is not exact, it is leaving aside the mass diference and the EM interaction; isn't that cheating a bit?

Certainly not.

Consider a hydrogen atom in ground state and an experiment to select one direction in space; the experiment is set up such that any direction may become this selected direction, but this choice is not made in the beginning. An example is switching on a B-field.

In the case of rotational symmetry you can discuss such experiments, in case of gauge theories you can't. For isospin symmetry you can introduce dynamics breaking the symmetry, for gauge symmetry you can't. You cannot measure the (local) phase related to gauge groups. There is no such experiment, not even in principle; there is no mathematical framework to describe this experiment, neither in theory, nor in practice.

 

[TrickyDicky]

Certainly not.

Consider a hydrogen atom in ground state and an experiment to select one direction in space; the experiment is set up such that any direction may become this selected direction, but this choice is not made in the beginning. An example is switching on a B-field.

In the case of rotational symmetry you can discuss such experiments,

in case of gauge theories you can't. For isospin symmetry you can introduce dynamics breaking the symmetry, for gauge symmetry you can't. You cannot measure the (local) phase related to gauge groups. There is no such experiment, not even in principal; there is no mathematical framework to describe this experiment, neither in theory, nor in practice.

Yes, that is how gauge symmetries are formally prescribed, although in practice I wouldn't use such definite terms, try switching on a B-field in the context of a Aharonov–Bohm effect kind of experiment and you'll measure a phase shift obviously related to the local EM gauge.
In my previous post I was rather questioning the validity of your argument wrt the rotation spacetime symmetry, if you had an observable completely invariant to rotations, based only on that observable it is quite obvious you couldn't perform an experiment to show symmetry breaking.