Theta-vacua and large gauge transformations
John C. Baez
under "large" gauge transformations (those not homotopic to the identity
- here one compactifies space to S^3) seems to be regarded as important.
More precisely (and I'm really just getting this from Huang's Quarks,
Leptons and Gauge Fields, and inviting corrections and clarification),
one can assume that the vacuum state psi(A) (as a function of the space
of connections) is invariant under gauge transforms homotopic to the
identity, but changes by a phase under large gauge transformations. OR
one can think of the theory as being truly gauge-invariant - but with a
different Lagrangian, with a "topological term" (2nd Chern class) thrown
in.
1) Is there any *mathematical* reason (as opposed to experimental) why
one believes that the vacuum of Yang-Mills theory shouldn't be invariant
under large gauge transformations?
2) One can cook up an operator that changes by an integer under large
gauge transformations - really just the Chern-Simons class. Has anyone
looked at this operator from the point of view of loop variables (i.e.,
worked out the commutators)?
l...@landau.phys.washington.edu
"John C. Baez" <jb...@BOURBAKI.MIT.EDU> writes:
>The idea that the vacuum state of a gauge theory needn't be invariant
>under "large" gauge transformations (those not homotopic to the identity
>- here one compactifies space to S^3) seems to be regarded as important.
>More precisely, one can assume that the vacuum state psi(A)
>(as a function of the space
>of connections) is invariant under gauge transforms homotopic to the
>identity, but changes by a phase under large gauge transformations. OR
>one can think of the theory as being truly gauge-invariant - but with a
>different Lagrangian, with a "topological term" (2nd Chern class) thrown in.
>1) Is there any *mathematical* reason (as opposed to experimental) why
>one believes that the vacuum of Yang-Mills theory shouldn't be invariant
>under large gauge transformations?
There is no *mathematical* reason why "the" vacuum should be invariant
under large gauge transformations - just as for a particle moving in a
periodic potential, for each choice of theta (or the eigenvalue of large
gauge transformations - analogous to the lattice momentum) there is a
different ground state. The only thing which is special to the field
theory is that no local operator can have a non-zero matrix element
connecting different theta-vacua. Consequently, each theta-vacua
describes a distinct physical theory (with properties depending on the
value of theta). [Technically, this break-up of the "big" Hilbert space
containing all states with all values of theta is the result of a
superselection rule.]
The only constraints on the value of theta come from experiment -
and imply that theta must be extremely close to zero.
N.B. - While theta has no dynamics, and is an adjustable parameter in
pure Yang-Mills theory, the situation is more complicated in axionic
models (in which additional degrees of freedom produce real dynamics
for theta).
>2) One can cook up an operator that changes by an integer under large
>gauge transformations - really just the Chern-Simons class. Has anyone
>looked at this operator from the point of view of loop variables (i.e.,
>worked out the commutators)?
I'm not sure just what you mean, but there has been a great deal of work
related to the Chern-Simons number [renormalization, effects when
coupled to a "chemical potential", connections with anomalous particle
production, ...]. Much of this work has been concerned with the electroweak
theory (as opposed to QCD) where the SU(2)_L Chern-Simons number is connected
with anomalous violation of baryon number conservation.
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Laurence G. Yaffe Internet: l...@newton.phys.washington.edu
University of Washington Bitnet: ya...@uwaphast.bitnet