Internal (gauge) symmetries and spacetime symmetries

[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.

 

[strangerep]

What exactly is abstract space and (I guess) non-abstract?

I'll let you in on a secret: they're all abstract. But don't tell anyone. :-)

It's curious how this subject tends to generate more heat than I would have expected.

Rotations act on space and they act on the wave functions as well. Does that make them both physical and unphysical?

No, it just means that the abstract group $SO(3)$ is being represented as operators on different vector spaces.

 

[strangerep]

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.

Not sure what you mean by "obviously related" here. The phase difference observed in an A-B setup does not depend on local EM gauge. The closed line integral therein is gauge-invariant.

 

[strangerep]

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

"Many loopholes"?? I found only a reference to the use of SUSY algebras, and a reference to 2D systems which admit different spin-statistics rules. The latter is not relevant for 4D field theory of elementary particles, and recent LHC results now make the former seem very unlikely.

 

[tom.stoer]

Yes, that is how gauge symmetries are formally prescribed, although in practice I wouldn't use such definite terms, ...

But you should ;-)

... 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.

The Aharonov-Bohm effect is due to a non-trivial global structure of the gauge fibre bundle; it does not measure a local phase as can be seen by varying either the path or the gauge infinitesimally; only changing the global structure i.e. the winding number (or the flux = the B-field) changes the Aharonov-Bohm phase.

... 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.

Correct, but usually we are talking about a complete set observables. Even this complete set of observables cannot detect any gauge structure b/c all observables must be gauge invariant i.e. must commute with the generators of gauge transformations.

 

[tom.stoer]

Regarding a rigorous formulation, propositions and loopholes of the Coleman-Mandula theorem I recommend http://arxiv.org/pdf/hep-th/9605147v1.pdf

One loophole could be the existence of the S-matrix itself; it's by no means clear that the S-matrix can be defined for the full theory, not just on special regimes, spacetime backgrounds etc.

 

[strangerep]

[...] what Alain Connes is trying to with his noncommutative SM. [...]

IIRC, Connes' model ran into conflict with experiment back in 2008 when his favored Higgs mass prediction was ruled out:

http://noncommutativegeometry.blogspot.com.au/2008/08/irony.html

He seems not to have given up all hope however. In response to the recent CERN Higgs announcement, he posted this:

http://noncommutativegeometry.blogspot.com.au/2012/07/habemus-higgs.html

but doesn't reconcile the enhanced models (with incorporate SUSY) with the LHC exclusion results regarding SUSY models.

 

[martinbn]

No, it just means that the abstract group $SO(3)$ is being represented as operators on different vector spaces.

What do you mean "no"! That's what I am saying.

 

[TrickyDicky]

Not sure what you mean by "obviously related" here. The phase difference observed in an A-B setup does not depend on local EM gauge. The closed line integral therein is gauge-invariant.

The Aharonov-Bohm effect is due to a non-trivial global structure of the gauge fibre bundle; it does not measure a local phase as can be seen by varying either the path or the gauge infinitesimally; only changing the global structure i.e. the winding number (or the flux = the B-field) changes the Aharonov-Bohm phase.

Ok, so you are making a distinction between global and local gauge, but they are both gauge symmetries, right? How are they related? This is what I meant, that the global EM gauge should be related to the local EM gauge. Surely the phase shift measured is the global one, that is because mathematically the local one can't be measured by definition.

 

[tom.stoer]

The difference is the global structure auf the fibre bundle. In trivial cases this is the direct product of 4-spacetime * gauge group. Locally this is always true, at each spacetime point you have a copy of the gauge group. But globally this can be complicated and there is a non-trivial topology of the fibre bundle. In the case of the Aharonov-Bohm effect this is due to the string-like defect in 3-space. Now there exists a winding number structure on the fibre bundle which means that gauge transformations fall into categories labelled by the winding number. Field configurations can now be related by small gauge transformations or by large ones; for large ones they be transformed continuously to each other by local transformations w/o violating the winding number.

Think about a circle S¹ and a gauge group U(1) which is usually written as

$$U[\theta]= \exp(2\pi i \theta(x))$$

with gauge function θ(x)

In contrast the the real line x is now an angle variable in the intervall [0,1]; the gauge group contains all gauge transformations which are periodic, i.e.

$$U[\theta(x+2\pi)] = U[\theta(x)]$$

Other functions θ(x) would violate the topology of fields living on S¹. Therfeore the (topology of the) gauge group is different for x living on a circle and x living on the real line!

Now think about a family of gauge functions

$$\theta_s(x) = s\,\theta(x)$$

For small θ(x) this is an allowed family, i.e. every θ_s(x) is a valid gauge function.

But now look at the simple function

$$\theta(x) = x$$

with
$$\theta(0) = 0$$
$$\theta(1) = 1$$

$$U[\theta(x=0)]=\exp(2\pi i \theta(0)) = \exp(0) = 1 = \exp(2\pi i) = \exp(2\pi i \theta(1)) = U[\theta(x=1)]$$

U is still periodic but θ isn't!

Nevertheless U is a valid gauge transformation b/c it respects the periodicity of S¹.

But in that case the family

$$\theta_s(x) = s\,\theta(x) = sx$$

is no longer a valid family of gauge functions; only s=0, s=±1, s=±2, ... are allowed!

These are exactly the representatives of different classes of gauge functions, one for each winding number n = 0, ±1, ±2, ... In general the functions

$$\theta_n(x) = nx$$

are valid functions for integer n: going round the S¹ circle from x=0 to x=1 exactly once the gauge transformation U[θ_n] runs around the circle n times:

$$U[\theta_n]= \exp(2\pi i \theta_n(x)) = \exp(2\pi i n x)$$

For fixed winding number n small fluctuations of the gauge functions are still llowed, i.e. if you have small functions covering the n=0 case, an arbitrary gauge transformation can be written as

$$U[\theta_n]= \exp(2\pi i \theta_n(x)) = \exp(2\pi i (nx + \tilde{\theta}(x))$$

where the second function is always a small one with winding number 0.

In the case of the Aharonov-Bohm effect the winding number becomes a physical observable whereas the small gauge transformations are still unphysical i.e. non-observable.

 

[TrickyDicky]

Thanks for all that info, Tom.
Some parts are over my head but I think I get the picture of the topological defect explanation of the effect.
Now, let us consider an alternative, or complementary, take on this. Instead of going directly to the fiber bundle structure of the circle group we could think that actually what we perceive as U(1) is not just U(1) but a broader group symmetry that has it as subgroup and that under certain set ups behaves like a global defect. This is the kind of approach I was thinking of in the OP.

 

[tom.stoer]

But my impression was that out of this broader symmetry group you may want to derive both spacetime and internal symmetries. This is - as far as we know - not possible for rather general conditions.

As I said: the Coleman-Mandula-theorem is derived in the context of the S-matrix. What happens if I have a theory which does not allow for the definition of an S-matrix in all its regimes?

 

[strangerep]

What do you mean "no"! That's what I am saying.

My "no" was in answer to your question about whether "that makes [rotations] both physical and unphysical".

 

[strangerep]

[...] the Coleman-Mandula-theorem is derived in the context of the S-matrix. What happens if I have a theory which does not allow for the definition of an S-matrix in all its regimes?

What exactly do you mean by "regimes" here?

 

[TrickyDicky]

But my impression was that out of this broader symmetry group you may want to derive both spacetime and internal symmetries. This is - as far as we know - not possible for rather general conditions.

As I said: the Coleman-Mandula-theorem is derived in the context of the S-matrix. What happens if I have a theory which does not allow for the definition of an S-matrix in all its regimes?

Yes, then you can combine internal and spacetime symmetries in a non-trivial way.
But then you'd need a non-SM theory, without mass gap and with a true spontaneous symmetry breaking (unlike the Higgs mechanism of symmetry breaking), correct?

 

[tom.stoer]

I do not want to identify specific loopholes for a given framework (quantum field theory on spacetime, operators, S-matrix, ...) but I want to question the framework itself!

QCD formulated in it's fundamental d.o.f. has a "confinement-regime" where an S-matrix for asymptotic quarks and gluon states does not exist. So strictly speaking the framework of the Coleman-Mandula theorem does not apply for QCD as a whole. I guess the same could be true for strongly coupled gravity in a deep quantum gravity regime where an S-matrix (for gravitons) may become meaningless.

Formulating theories for emergent or discrete spacetime like (colored) spin networks may render the concept for "S-matrix on spacetime" meaningless. The question then is whether you can still use this S-matrix approach which is restrcited to a certain regime to derive a no-go theorem for the whole theory. I doubt that this will work and therefore I expect that many of these no-go theorems will cease to exist when identifying a fundamental theory of quantum gravity. Besides the Coleman-Mandula theorem the Weinberg–Witten theorem is another candidate to fail.

 

[TrickyDicky]

I do not want to identify specific loopholes for a given framework (quantum field theory on spacetime, operators, S-matrix, ...) but I want to question the framework itself!

Hey, didn't know you were so radical.

Formulating theories for emergent or discrete spacetime like (colored) spin networks may render the concept for "S-matrix on spacetime" meaningless. The question then is whether you can still use this S-matrix approach which is restrcited to a certain regime to derive a no-go theorem for the whole theory. I doubt that this will work and therefore I expect that many of these no-go theorems will cease to exist when identifying a fundamental theory of quantum gravity. Besides the Coleman-Mandula theorem the Weinberg–Witten theorem is another candidate to fail.

I share this view about no-go theorems.

 

[tom.stoer]

Hey, didn't know you were so radical.

depends on the enemy ...

 

[PhilDSP]

I doubt that this will work and therefore I expect that many of these no-go theorems will cease to exist when identifying a fundamental theory of quantum gravity.

Now I have a question bearing some level of radicalness :)

Is there an empirical reason to be led to believe that gravity is (or should be) quantized? Or is it principally a consistency in frameworks that one assumes should be accommodated?

 

[TrickyDicky]

Now I have a question bearing some level of radicalness :)

Is there an empirical reason to be led to believe that gravity is (or should be) quantized? Or is it principally a consistency in frameworks that one assumes should be accommodated?

I remember having read somewhere a physicist that claimed that gravity was already "quantized" in GR. I will try to find the reference because I can't remember what kind of arguments he used or what he meant.

 

[tom.stoer]

Is there an empirical reason to be led to believe that gravity is (or should be) quantized? Or is it principally a consistency in frameworks that one assumes should be accommodated?

There is no direct experimental evidence for quantum gravity, so I would say that the indications are mostly due to theoretical and consistency reasons; but there are a lot ...