In Minkowski space even-dim (say d+1 D) spacetime dimension, we can write fermion-field theory as the Lagrangian:
L=ˉψ(i∂̸−m)ψ+ˉψϕ1ψ+ˉψ(iγ5)ϕ2ψ
with Yukawa coupling to scalar term (with ϕ1) and pseudoscalar term (with ϕ2). Here ˉψ=ψ†γ0. This L is:
(i) Hermitian
(ii) invariant under Lorentz group SO(d,1).
In the Euclidean fermion field theory, there are many attempts in the literature to find an analytic continuation from Minkowski to Euclidean field theory, and to satisfy the conditions of:
(i) Hermitian
(ii) invariant under rotational group SO(d+1)
for example, in Steven Weinberg Vol II,
LE=ψ†(i∂̸E−m)ψ
or this PRL paper,
LE=ψ†iγE5(i∂̸E−m)ψ
in the case of this PRL paper, γE0=iγ5, γE5=−iγ0, and γEj=γEj for j=1,2,…
The clifford algebra: γμ,γν=2(+,−,−,−,…), then
γEμ,γEν=2(−,−,−,−,…)
with
xE=(tE,x1,x2,x3,…)
I wonder in the Euclidean fermion field theory:
(Q1) Time Reversal and CPT: how does the Time Reversal T work in Euclidean theory, say TE? Is time reversal TE the same as parity operator P then? Is TE unitary or anti-unitary? And whether there is CPT theorem still? If partity P and time reversal TE operators are (almost) the same meaning in Euclidean space, then how CPT theorem in Euclidean field theory version becomes?
(Q2) spin and spin-statistics:
In Minkowski signature, although spin state does not change under parity P, but the spin state changes under time reversal T - spin flips direction, (see, for example, Peskin and Schroeder QFT p.68), say in 4D (d+1=4),
ξs→(−iσ2)(ξs)∗≡ξ−s
Flip spin twice and measuring the spin eigenvalue by (→n⋅→σ),
(→n⋅→σ)ξs=sξs→(→n⋅→σ)(−iσ2)((−iσ2)(ξs)∗)∗=(→n⋅→σ)((−1)ξs)=s((−1)ξs)
will give the same eigenvalue of s, but the statefunction picks up a (-1) sign. This is related to spin-statistics theorem of fermions.
In Euclidean signature, whether there is a defined notion of spin? Does spin state change under parity P and time reversal TE? Is there any kind of spin-statistics theorem in Euclidean signature?
(Q3) pseudoscalar:
In Minkowski, we have a Hermitian pseudoscalar term LM,pseudo=ˉψ(iγ5)ψ,
with:
PLM,pseudoψP=−LM,pseudo
TLM,pseudoψT=−LM,pseudo
CLM,pseudoC=+LM,pseudo
In Euclidean space, is there a peudoscalar term LE,pseudo in the version of this Euclidean theory: LE=ψ†iγE5(i∂̸E−m)ψ, such that
LE,pseudo is Hermitian and SO(d+1) rotational invariance, and with
PLE,pseudoψP=−LE,pseudo
TELE,pseudoψTE=−LE,pseudo
CLE,pseudoC=+LE,pseudo
what is LE,pseudo=? in the case of 2D (d+1=2) and 4D (d+1=4)?
(Q4) mass gap and chirality:
In Minkowski theory, the mass gap is due to the term:
LM,mass=mˉψψ=m(ψ†LψR+ψ†RψL)
where ψL,ψR interpreted as massless limit of Left-, Right-handed Weyl spinors.
In Euclidean theory, however, in Steven Weinberg QFT II's theory, the mass term is written as mψ†ψ, does that mean
LE,mass=mψ†ψ=m(ψ†LψL+ψ†RψR)
the mass gap is generated from a forward scattering term( L→L,R→R)? (This is weird!) Or does the projection PL,PR and chirality is defined differently in Euclidean theory? How does this projection PL,PR reflect the group structure of
SO(4)≃SU(2)×SU(2)/Z2 in the case of 4D(d+1=4)?
This post imported from StackExchange Physics at 2014-06-04 11:36 (UCT), posted by SE-user Idear