# Time Reversal, CPT, spin-statistics, mass gap and chirality of Euclidean fermion field theory

In Minkowski space even-dim (say $d+1$ D) spacetime dimension, we can write fermion-field theory as the Lagrangian:

$$\mathcal{L}=\bar{\psi} (i\not \partial-m)\psi+ \bar{\psi} \phi_1 \psi+\bar{\psi} (i \gamma^5)\phi_2 \psi$$

with Yukawa coupling to scalar term (with $\phi_1$) and pseudoscalar term (with $\phi_2$). Here $\bar{\psi}=\psi^\dagger\gamma^0$. This $\mathcal{L}$ 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, $$\mathcal{L}_E= \psi^\dagger(i\not \partial_E-m)\psi$$

or this PRL paper,

$$\mathcal{L}_E= \psi^\dagger i \gamma^E_5(i\not \partial_E-m)\psi$$

in the case of this PRL paper, $\gamma^E_0=i \gamma_5$, $\gamma^E_5=-i \gamma_0$, and $\gamma^E_j=\gamma^E_j$ for $j=1,2,\dots$

The clifford algebra: ${\gamma_\mu,\gamma_\nu}=2(+,-,-,-,\dots)$, then $${\gamma^E_\mu,\gamma^E_\nu}=2(-,-,-,-,\dots)$$ with $x_E=(t_E,x_1,x_2,x_3,\dots)$

I wonder in the Euclidean fermion field theory:

(Q1) Time Reversal and CPT: how does the Time Reversal $T$ work in Euclidean theory, say $T_E$? Is time reversal $T_E$ the same as parity operator $P$ then? Is $T_E$ unitary or anti-unitary? And whether there is CPT theorem still? If partity $P$ and time reversal $T_E$ 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),

$$\xi^s \to (-i \sigma_2)(\xi^s)^* \equiv \xi^{-s}$$

Flip spin twice and measuring the spin eigenvalue by $(\vec{n} \cdot \vec{\sigma})$,

$$(\vec{n} \cdot \vec{\sigma}) \xi^s = s \xi^s \to \\ (\vec{n} \cdot \vec{\sigma}) (-i \sigma_2 )\big((-i \sigma_2 )(\xi^s)^*\big)^* = (\vec{n} \cdot \vec{\sigma}) (\mathbf{(-1)}\xi^s) = s (\mathbf{(-1)}\xi^s)$$

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 $T_E$? Is there any kind of spin-statistics theorem in Euclidean signature?

(Q3) pseudoscalar:

In Minkowski, we have a Hermitian pseudoscalar term $\mathcal{L}_{M,pseudo}=\bar{\psi} (i \gamma^5) \psi$,

with:

$$P \mathcal{L}_{M,pseudo} \psi P= -\mathcal{L}_{M,pseudo}$$

$$T \mathcal{L}_{M,pseudo} \psi T= -\mathcal{L}_{M,pseudo}$$

$$C \mathcal{L}_{M,pseudo} C= +\mathcal{L}_{M,pseudo}$$

In Euclidean space, is there a peudoscalar term $\mathcal{L}_{E,pseudo}$ in the version of this Euclidean theory: $\mathcal{L}_E=\psi^\dagger i \gamma^E_5(i\not \partial_E-m)\psi$, such that $\mathcal{L}_{E,pseudo}$ is Hermitian and SO(d+1) rotational invariance, and with

$$P \mathcal{L}_{E,pseudo} \psi P= -\mathcal{L}_{E,pseudo}$$

$$T_E \mathcal{L}_{E,pseudo} \psi T_E= -\mathcal{L}_{E,pseudo}$$

$$C \mathcal{L}_{E,pseudo} C= +\mathcal{L}_{E,pseudo}$$

what is $\mathcal{L}_{E,pseudo}=?$ 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:

$$\mathcal{L}_{M,mass}=m \bar{\psi}\psi= m (\psi_L^\dagger \psi_R+\psi_R^\dagger \psi_L)$$

where $\psi_L,\psi_R$ 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 $m \psi^\dagger \psi$, does that mean

$$\mathcal{L}_{E,mass}=m \psi^\dagger \psi=m (\psi_L^\dagger \psi_L+\psi_R^\dagger \psi_R)$$

the mass gap is generated from a forward scattering term( $L\to L, R\to R$)? (This is weird!) Or does the projection $P_L, P_R$ and chirality is defined differently in Euclidean theory? How does this projection $P_L, P_R$ reflect the group structure of

$$SO(4) \simeq SU(2) \times SU(2)/Z_2 \text{ in the case of 4D} (d+1=4)?$$