Mathematical Quantum Field Theory - Gauge symmetries - Comments

[Urs Schreiber]

Greg Bernhardt submitted a new PF Insights post

Mathematical Quantum Field Theory - Gauge symmetries

Continue reading the Original PF Insights Post.

 

[Jimster41]

Thanks for taking the time to write this and try to teach this stuff to such a broad audience.

So, just to confirm I actually got some of the first section. The existence of the non-trivial infinitesimal symmetries (the assumptions about space-time that are required to support them) present a problem for the notion of the co-variant derivative (in other words the existence of the Cevi-Levita connection)? Is that roughly following? If not where did I go off the rails, if you can possibly say?

If I am not getting that completely wrong then I am excited because you seem to be talking about the old "how long can the infinitely rigid metal rod sticking off the Earth be" - problem. In other words how can the space-time field support both continuous infinitesimal symmetry (the rod's integrity) and stretchy space-time between accelerated frames. I am totally curious what the proposed solution is - that doesn't abolish or somehow dilute infinitesimal space-time field symmetry.

and will continue to try to read...even though the math looks totally

I enjoy getting lost on the nLab site semi-regularly.

[Edit] now I think I am probably totally getting it wrong.

 

[Urs Schreiber]

The existence of the non-trivial infinitesimal symmetries [...] present a problem for the notion of the co-variant derivative (in other words the existence of the Cevi-Levita connection)?

Good that you ask a question!
As you already suspected, the answer to this question is: No. But that's what questions are for.

It sounds like your question originates in conflating the term "covariant phase space" with the term "covariant derivative". These are completely different concepts. The adjective "covariant" is used for lots of things in lots of situations. Not all concepts that come with the word "covariant" are related.

For the record, I'll say again what the issue is regarding the covariant phase space in the case of gauge theories:

If gauge symmetries are present, in the sense discussed in the entry, then specifying initial value data (i.e "canonical coordinates and canonical momenta") for field histories on a spatial slice (meaning roughly "at some instant of time") cannot uniquely fix the field histories in the past and future. This is because with anyone field history that has the given initial values, also any gauge transformation of that field history away from that spatial slice will have the same.

Now the "covariant phase space", if it exists, is the space of all field histories that solve the equations of motion, parameterized by the initial value data of these field histories on good spatial slices (Cauchy surfaces). If this exists, it comes equipped with a Poisson bracket, and this is what controls the quantum theory ("canonical commutation relation between coordinates and momenta"). If the covariant phase space does not exist, then there is no Poisson bracket, and hence no quantum theory.

In the presence of gauge symmetries, we just saw that initial value data for field histories on spatial slices does not serve as parameterization for all on-shell field histories, and hence in this case the covariant phase space does not exist, and hence the quantum theory does not exist.

For this reason, if there are gauge symmetries than one needs to re-think before one can pass to the quantum theory. Namely one has to find a different field theory which

a) does have a covariant phase space;

b) is still equivalent to the original field theory in a suitable sense.

This re-thinking is called BV-BRST gauge fixing, discussed in chapter 12. Gauge fixing.

 

[dextercioby]

Urs, when you name some formalism as ~BV~, which of the very many papers written by Batalin and Fradkin in the 1980s do you mean exactly? I understand you will come up at the end (or so I hope) with a full reference/further-reading list. Will you include the most relevant papers (to your lectures) by Fradkin, Fradkina and Batalin on the BRST topic?

 

[Jimster41]

It sounds like your question originates in conflating the term "covariant phase space" with the term "covariant derivative". These are completely different concepts. The adjective "covariant" is used for lots of things in lots of situations. Not all concepts that come with the word "covariant" are related.

Thanks, Urs.
A couple of further clarification would be appreciated.
Does the existence of a covariant derivative depend on the existence of a covariant phase space?
A Lorentz transformation is not a gauge transformation, correct? I think I am conflating those two things also.

 

[Urs Schreiber]

Urs, when you name some formalism as ~BV~, which of the very many papers written by Batalin and Fradkin in the 1980s do you mean exactly? I understand you will come up at the end (or so I hope) with a full reference/further-reading list. Will you include the most relevant papers (to your lectures) by Fradkin, Fradkina and Batalin on the BRST topic?

I have citations inside each chapter, attached to the definitions and/or propositons that I am discussing. I think what I provide is pretty self-contained, but let me know precise points where you'd like to see more comments on the relation to the literature.

For the Lie-algebroid perspective on BRST that I develop in this chapter 10. Gauge symmetries, you may compare to Barnich 10.

For the local BV-BRST complex laid out in chapter 11. Reduced phase I am following Barnich-Brandt-Henneaux 00.

For the BV-gauge fixing developed in 12. Gauge fixing I took clues from Fredenhagen-Rejzner 11a.

For the free quantum BV in 13. Free quantum field and the interacting quantum BV in 15. Interacting quantum fields (upcoming) I am following Fredenhagen-Rejzner 11b, Rejzner 11, which in turn is taking clues from Hollands 07.

For general discussion I recommend Henneaux 90 (only that wherever Henneaux says "path integral" you should ignore it and instead use the rigorous construction in 13. Free quantum fields, and 15. Interacting quantum fields (upcoming), following Fredenhagen-Rejzner 11b, Rejzner 11.

 

[Urs Schreiber]

Does the existence of a covariant derivative depend on the existence of a covariant phase space?

No, these are two completely independent concepts.

1) A covariant phase space is a space of solutions to equations of motion in field theory, equipped with a symplectic form (roughly: a way to speak about "canonical coordinates" and "canonical momenta".)

2) A covariant derivative is a structure on a collection of spaces that vary smoothly over a base space which provides a rule for how to lift paths in the base space to paths in the collection of these spaces.

The covariant derivative in 2) happens to play a role in the description of gauge fields and of the field of gravity. After gauge fixing gauge field theories and/or gravity, these theories have a phase space as in 1). In this way both 1) and 2) play a role in physics. But as concepts they are not logically related.

 

[Urs Schreiber]

I understand you will come up at the end (or so I hope) with a full reference/further-reading list.

Okay, I have now included something in the first article under References.

 

[Duong]

1. In defining the action Lie algebroids, we repeatedly use the notation ## X/\mathfrak{g} ##. Does it in any way suggest here we're dealing with a space of orbits, say, ##X/G##?

2. In defining infinitesimal gauge symmetries, we let the functions ##R^{a \mu_1 ... \mu_k}_\alpha \in C^\infty (J^\infty_\Sigma(E))## be independent of the spacetime coordinates ##x^\nu##. As I understand, this fact is used in proving the Noether identities as then ## R^{a \mu_1 ... \mu_k}_\alpha ## play no roles in the integration by part process. But on what basis could we suppose ## R^{a \mu_1 ... \mu_k}_\alpha ## to be so?

 

[Urs Schreiber]

1. In defining the action Lie algebroids, we repeatedly use the notation ## X/\mathfrak{g} ##. Does it in any way suggest here we're dealing with a space of orbits, say, ##X/G##?

Yes, it's a "homotopy quotient", hence a space of "orbits up to homotopy". I am not explaining the relevant homotopy-theory in the QFT notes, since it leads too far. But in chapter 11. Reduced phase space it is spelled out in example 11.2 that ##X/\mathfrak{g}## does behave like a space of gauge orbits in that smooth functions on it are equivalently the gauge-invariant functions on ##X##.

 

[Urs Schreiber]

2. In defining infinitesimal gauge symmetries, we let the functions ##R^{a \mu_1 ... \mu_k}_\alpha \in C^\infty (J^\infty_\Sigma(E))## be independent of the spacetime coordinates ##x^\nu##. As I understand, this fact is used in proving the Noether identities as then ## R^{a \mu_1 ... \mu_k}_\alpha ## play no roles in the integration by part process. But on what basis could we suppose ## R^{a \mu_1 ... \mu_k}_\alpha ## to be so?

Woops. No, this was a bad typo on my part. The ##R##s must be inside the differentiation! I have fixed it (prop. 10.9) You can see this in action in the differential of the local BV-BRST complex in example 11.21 (where it was typed correctly all along). Thanks for catching this. Sorry for the bad typo.

 

[Duong]

Thanks! I guess it would help if you could include ##x^\nu## in the line about ##R_\alpha^{a\mu_1 ... \mu_k} ## in the definition of infinitesimal gauge symmetries. By now only the dependence on the field and its jet order coordinates is explicit.

Aha! I do see the quotient structure on the function level now. Very nice. What is a homotopy quotient remains for me to learn.

A few more questions about the proof of Noether's identities:

1. I think to be precise we need to include a factor ##dvol_\Sigma## to the outside of the summation term of equation (3).

2. I am fine with equation (3) if here we already gauge parameterized the infinitesimal symmetry by a section ##\epsilon \in \Gamma_\Sigma(\mathcal{G}) ##, as then the integration by parts is from ordinary calculus. But are we resorting to such ##\epsilon## here? If not, then what is the general statement of the jet-level integration by parts?

One question on the local BRST complex :

3. How to define the operation of ##d_{CE}## on ## c^\alpha_{,\mu_1...\mu_k}##?
It is said in the text that this could be done via the anti-commutativity of ##s_{BRST}## with the horizontal derivative ##d##. But how is it done? I am not even sure what is ##d c^\alpha_{,\mu_1...\mu_k} ##.

 

[Urs Schreiber]

it would help if you could include ##x^\nu## in the line about ##R_\alpha^{a\mu_1 ... \mu_k} ## in the definition of infinitesimal gauge symmetries.

Okay, I can add that. Bur notice that this is not really the issue that you were highlighting. Even without explicit dependence on the spacetime coordinates, the spacetime-derivatives of the ##R## are in general non-trivial, since the total spacetime derivative alse differentiates with respect to all the field coordinates, by equation (27) in chapter 4. Field variations .

What is a homotopy quotient remains for me to learn.

For a very gentle exposition see Higher Structures in Mathematics and Physics. For full background you need Introduction to Homotopy Theory. But this is not crucial for the MQFT notes at the moment. It becomes important when stepping back and understanding the bigger picture of QFT, such as homotopical AQFT.

we need to include a factor ##dvol_\Sigma## to the outside of the summation term of equation (3).

That's supposed to be absorbed by the boldface ##\mathbf{L}##. I am trying to consistently write

$$ \mathbf{L} = L \, dvol_\Sigma$$

where thus ##\mathbf{L}## is the "Lagrangian density" while its coefficient ##L## with respect to the given spacetime volume form is the "Lagrangian function".

But let me know if you spot places where this convention is not kept consistently.

 

[Urs Schreiber]

what is the general statement of the jet-level integration by parts?

It just means the observation that we may throw over the spacetime (horizontal) derivative from one factor to the other by picking up a total spacetime derivative. I have made this more explicit now by giving it a numbered environment, now example 7.37. Let me know if this helps.

I am not even sure what is ##d c^\alpha_{,\mu_1...\mu_k} ##.

The point is that the ghost fields are indeed fields (they have a field bundle, namely ##\mathcal{G}[1] \overset{gb}{\to} \Sigma##), hence are subject to the precise same variational calculus as the ordinary fields. By the definition of the total spacetime derivative (equation (27)), we have

$$ d c^\alpha_{,\mu_1...\mu_k} = c^\alpha_{,\mu_1...\mu_k \mu_{k+1} } \mathbf{d} x^{\mu_{k+1}}$$

 

[Duong]

Even without explicit dependence on the spacetime coordinates, the spacetime-derivatives of the ##R## are in general non-trivial, since the total spacetime derivative alse differentiates with respect to all the field coordinates

I see this now. Thank you -- also for the pointers towards homotopy theory.

That's supposed to be absorbed by the boldface ##\mathbf{L}##.

I assume then ##\frac{d^k}{dx^{\mu_1}...dx^{\mu_k}} \left( R \frac{\delta_{EL}\mathbf{L}}{\delta\phi} \right)## is a notation for ##\frac{d^k}{dx^{\mu_1}...dx^{\mu_k}} \left( R \frac{\delta_{EL}L}{\delta\phi} \right) dvol_\Sigma## ?

 

[Duong]

It just means the observation that we may throw over the spacetime (horizontal) derivative from one factor to the other by picking up a total spacetime derivative.

I am still uncomfortable on this. I suppose we're using ##d c^\alpha_{,\mu_1...\mu_k} = c^\alpha_{,\mu_1...\mu_k \mu_{k+1} } \mathbf{d} x^{\mu_{k+1}} ## already here and exchanging the horizontal derivatives on ##c_{,\mu_1...\mu_k} ## and on a spacetime volume form. But isn't the jet bundle on which we are probing whether ##R## is its infinitesimal symmetry is just ## J^\infty_\Sigma (E) ##, where the ##c##'s are pure real numbers (once we fix a point on ##J^\infty_\Sigma(\mathcal{G}) ## and so fix a would-be-infinitesimal symmetry) and so the notion of ##dc## does not make sense?

 

[Urs Schreiber]

I am still uncomfortable on this. I suppose we're using ##d c^\alpha_{,\mu_1...\mu_k} = c^\alpha_{,\mu_1...\mu_k \mu_{k+1} } \mathbf{d} x^{\mu_{k+1}} ## already here and exchanging the horizontal derivatives on ##c_{,\mu_1...\mu_k} ## and on a spacetime volume form. But isn't the jet bundle on which we are probing whether ##R## is its infinitesimal symmetry is just ## J^\infty_\Sigma (E) ##, where the ##c##'s are pure real numbers (once we fix a point on ##J^\infty_\Sigma(\mathcal{G}) ## and so fix a would-be-infinitesimal symmetry) and so the notion of ##dc## does not make sense?

No the jet bundle on which this applies is that of ##E \times_\Sigma \mathcal{G}## (if we regard the gauge parameters ##\epsilon## in degree 0) or, without change of the conclusion, is ##E \times_\Sigma \mathcal{G}[1]## if we replace the gauge parameters ##\epsilon## by the ghost field ##c##.

I'll try to add a clarification to the text at this point.

 

[Urs Schreiber]

I assume then ##\frac{d^k}{dx^{\mu_1}...dx^{\mu_k}} \left( R \frac{\delta_{EL}\mathbf{L}}{\delta\phi} \right)## is a notation for ##\frac{d^k}{dx^{\mu_1}...dx^{\mu_k}} \left( R \frac{\delta_{EL}L}{\delta\phi} \right) dvol_\Sigma## ?

Yes! Sorry, I thought that's clear. I'll make this more explicit in the notes.