280
Appendix F: U(1) Problem Solved by Gauge Group Topology
it gives
β
λ
(V n ) = e
−i2nλ V n ,
the strict analog of Eq. (F.26).
A gauge invariant ground state ω 0 defines a non-regular representation of A W
ω 0 (U (α) V (β)) = δ α,0 ω 0 (V (β)).
The representatives of the operators V n , for simplicity still denoted by V n , correspond
to the T n introduced in Proposition F.4, and their spectrum {e
i2nθ
, θ ∈ [ 0, π)} provides the θ angle which labels the factorial representation of the observable algebra.
Exactly as for the solution of the U (1) problem in QCD, the “chiral symmetry” β
λ
may be defined by the unitary operators U (λ/π), which are not weakly continuous
in the parameter λ, as the unitary operators V
5
R (λ) of eqs. (F.7), (F.22); hence, the
corresponding local generator of infinitesimal transformations does not exist and one
cannot write the symmetry breaking Ward identities at the basis of the argument of
Goldstone theorem (evasion of Goldstone theorem).
References Appendix
P.W. Anderson, Phys. Rev. 112, 1900 (1958)
W.A. Bardeen, Nucl. Phys. B 246, (1974)
J. Bardeen, L. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)
B. Booß-Bavnbeck, G. Morchio, F. Strocchi, K.P. Wojciechowski, J. Geom. Phys. 22, 219 (1997)
K. Brading, H.R. Brown, Noether’s Theorems and Gauge Symmetries. arXiv:hep-th/0009058
K. Brading, H.R. Brown, Symmetries and Noether’s Theorems, in Symmetries in Physics: Philosophical Reflections, K. Brading, E. Castellani ed., (Cambridge University Press, Cambridge,
2003a)
K. Brading, E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections (Cambridge
University Press, Cambridge, 2003b)
A. Cintio, G. Morchio, J. Math. Phys 50, 042102 (2009)
S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985)
R.J. Crewther, Chiral properties of quantum chromodynamics, in Field Theoretical Methods in
Particle Physics, W. Rühl ed., (Reidel 1980), pp. 529–590
G.F. De Angelis, D. De Falco, F. Guerra, Phys. Rev. D 17, 1624 (1978)
G. De Palma, F. Strocchi, Ann. Phys. 336, 112 (2013)
G. de Rham, Differential Manifolds (Springer, Berlin, 1984)
D.A. Dubin, G.L. Sewell, J. Math. Phys. 11, 2990 (1979)
S. Elitzur, Phys. Rev. D 2, 3978 (1975)
Appendix F: U(1) Problem Solved by Gauge Group Topology
it gives
β
λ
(V n ) = e
−i2nλ V n ,
the strict analog of Eq. (F.26).
A gauge invariant ground state ω 0 defines a non-regular representation of A W
ω 0 (U (α) V (β)) = δ α,0 ω 0 (V (β)).
The representatives of the operators V n , for simplicity still denoted by V n , correspond
to the T n introduced in Proposition F.4, and their spectrum {e
i2nθ
, θ ∈ [ 0, π)} provides the θ angle which labels the factorial representation of the observable algebra.
Exactly as for the solution of the U (1) problem in QCD, the “chiral symmetry” β
λ
may be defined by the unitary operators U (λ/π), which are not weakly continuous
in the parameter λ, as the unitary operators V
5
R (λ) of eqs. (F.7), (F.22); hence, the
corresponding local generator of infinitesimal transformations does not exist and one
cannot write the symmetry breaking Ward identities at the basis of the argument of
Goldstone theorem (evasion of Goldstone theorem).
References Appendix
P.W. Anderson, Phys. Rev. 112, 1900 (1958)
W.A. Bardeen, Nucl. Phys. B 246, (1974)
J. Bardeen, L. Cooper, J.R. Schrieffer, Phys. Rev. 108, 1175 (1957)
B. Booß-Bavnbeck, G. Morchio, F. Strocchi, K.P. Wojciechowski, J. Geom. Phys. 22, 219 (1997)
K. Brading, H.R. Brown, Noether’s Theorems and Gauge Symmetries. arXiv:hep-th/0009058
K. Brading, H.R. Brown, Symmetries and Noether’s Theorems, in Symmetries in Physics: Philosophical Reflections, K. Brading, E. Castellani ed., (Cambridge University Press, Cambridge,
2003a)
K. Brading, E. Castellani (eds.), Symmetries in Physics: Philosophical Reflections (Cambridge
University Press, Cambridge, 2003b)
A. Cintio, G. Morchio, J. Math. Phys 50, 042102 (2009)
S. Coleman, Aspects of Symmetry (Cambridge University Press, Cambridge, 1985)
R.J. Crewther, Chiral properties of quantum chromodynamics, in Field Theoretical Methods in
Particle Physics, W. Rühl ed., (Reidel 1980), pp. 529–590
G.F. De Angelis, D. De Falco, F. Guerra, Phys. Rev. D 17, 1624 (1978)
G. De Palma, F. Strocchi, Ann. Phys. 336, 112 (2013)
G. de Rham, Differential Manifolds (Springer, Berlin, 1984)
D.A. Dubin, G.L. Sewell, J. Math. Phys. 11, 2990 (1979)
S. Elitzur, Phys. Rev. D 2, 3978 (1975)