Appendix F: U(1) Problem Solved by Gauge Group Topology
265
by the canonical equal time (anti-)commutators and yields the time-independent
infinitesimal chiral transformations δ
5 , (α (t) (x 0 ) ≡ α(x 0 − t))
α
t
δ
5
(ψ(x, 0)) = i lim
R
[ J
5
0 ( f R α (t) ), ψ(x, t) ] = i lim
R
[ J
5
0 ( f R α), ψ(x, t) ] =
= δ
5
α
t
(ψ(x, 0)).
(F.5)
This is in agreement with the perturbative analysis in local renormalizable
gauges
46 and one may conclude that J
5
0 generates the infinitesimal time-independent
chiral transformations δ
5 of Eq. (F.2).
47
Then, with the notations used in Chap. 27,
δ
5
ψ(h) = i lim
R
[ J
5
0 ( f R , α), ψ(h) ], δ
5 A
a
μ = 0.
(F.6)
Under general conditions on the local algebra generated by the local fields ψ, A
a
μ ,
one may consider the one-parameter group of unitary operators V
5
R (λ), (formally the
exponentials e
iλ J
5
0 ( f R α) ) and, by Eq. (F.6), obtain
β
λ
(F) = lim
R→∞
V
5
R (λ) F V
5
R (−λ), ∀F ∈ F,
(F.7)
where, by locality, the limit is reached for finite R. The conclusion is that the timeindependent chiral transformations are (algebraic) symmetries of the local field algebra F and, in particular, of its observable subalgebra F obs , a result which is independent of the gauge fixing and of the corresponding (gauge dependent) field algebra in
which F obs is embedded.
This disagrees with G. ’t Hooft statement that chiral symmetry does not exist as a
group of time-independent transformations of the observables; the arguments are that
“chiral symmetry is explicitly broken by instantons” and/or that it is a “fictitious” or
a “phoney symmetry” which exists only by using an “artificial huge” Hilbert space,
which contains all the chirally transformed states.
48
The gauge dependence of the unitary operators V
5
R (λ) does not invalidate Eq.
(F.7), since they are merely instrumental for the definition of the chiral transformations β
λ on the observable fields. It looks short sighted to blame on the fact that J
5
μ or
the (better behaved) exponentials V
5
R (λ) are gauge dependent not observable operators; such a point of view would in fact deny the very existence of the non-abelian
46 W.A. Bardeen, Nucl. Phys. B, 246 (1974).
47 At equal times, the commutator of j 5
0 formally gives the infinitesimal chiral transformations, but
for (infinite) renormalization constants due to vacuum polarization effects. Only with the addition
of the term K 5
μ , thanks to the non-renormalization theorem for local conserved currents, one gets rid
of the infinite renormalization constants and obtain the extension of the generation to unequal times
(K 5
0 involves only the spatial components A a
i of the vector potential and its equal time canonical
commutators with the fermion fields and with A a
μ formally vanish).
48 G. t’ Hooft, How instantons solve the U (1) problem, Physics Reports, 142, 357 (1986).
265
by the canonical equal time (anti-)commutators and yields the time-independent
infinitesimal chiral transformations δ
5 , (α (t) (x 0 ) ≡ α(x 0 − t))
α
t
δ
5
(ψ(x, 0)) = i lim
R
[ J
5
0 ( f R α (t) ), ψ(x, t) ] = i lim
R
[ J
5
0 ( f R α), ψ(x, t) ] =
= δ
5
α
t
(ψ(x, 0)).
(F.5)
This is in agreement with the perturbative analysis in local renormalizable
gauges
46 and one may conclude that J
5
0 generates the infinitesimal time-independent
chiral transformations δ
5 of Eq. (F.2).
47
Then, with the notations used in Chap. 27,
δ
5
ψ(h) = i lim
R
[ J
5
0 ( f R , α), ψ(h) ], δ
5 A
a
μ = 0.
(F.6)
Under general conditions on the local algebra generated by the local fields ψ, A
a
μ ,
one may consider the one-parameter group of unitary operators V
5
R (λ), (formally the
exponentials e
iλ J
5
0 ( f R α) ) and, by Eq. (F.6), obtain
β
λ
(F) = lim
R→∞
V
5
R (λ) F V
5
R (−λ), ∀F ∈ F,
(F.7)
where, by locality, the limit is reached for finite R. The conclusion is that the timeindependent chiral transformations are (algebraic) symmetries of the local field algebra F and, in particular, of its observable subalgebra F obs , a result which is independent of the gauge fixing and of the corresponding (gauge dependent) field algebra in
which F obs is embedded.
This disagrees with G. ’t Hooft statement that chiral symmetry does not exist as a
group of time-independent transformations of the observables; the arguments are that
“chiral symmetry is explicitly broken by instantons” and/or that it is a “fictitious” or
a “phoney symmetry” which exists only by using an “artificial huge” Hilbert space,
which contains all the chirally transformed states.
48
The gauge dependence of the unitary operators V
5
R (λ) does not invalidate Eq.
(F.7), since they are merely instrumental for the definition of the chiral transformations β
λ on the observable fields. It looks short sighted to blame on the fact that J
5
μ or
the (better behaved) exponentials V
5
R (λ) are gauge dependent not observable operators; such a point of view would in fact deny the very existence of the non-abelian
46 W.A. Bardeen, Nucl. Phys. B, 246 (1974).
47 At equal times, the commutator of j 5
0 formally gives the infinitesimal chiral transformations, but
for (infinite) renormalization constants due to vacuum polarization effects. Only with the addition
of the term K 5
μ , thanks to the non-renormalization theorem for local conserved currents, one gets rid
of the infinite renormalization constants and obtain the extension of the generation to unequal times
(K 5
0 involves only the spatial components A a
i of the vector potential and its equal time canonical
commutators with the fermion fields and with A a
μ formally vanish).
48 G. t’ Hooft, How instantons solve the U (1) problem, Physics Reports, 142, 357 (1986).