Discrete symmetries and the propagator approach to coupled fermions in Quantum Field Theory. Generalities: The case of a single fermion–antifermion pair

doi:10.1016/j.aop.2010.05.016

Annals of Physics

Volume 325, Issue 10, October 2010, Pages 2041-2074

https://doi.org/10.1016/j.aop.2010.05.016 Get rights and content

Abstract

Starting from Wigner’s symmetry representation theorem, we give a general account of discrete symmetries (parity P, charge conjugation C, time-reversal T), focusing on fermions in Quantum Field Theory. We provide the rules of transformation of Weyl spinors, both at the classical level (grassmanian wave functions) and quantum level (operators). Making use of Wightman’s definition of invariance, we outline ambiguities linked to the notion of classical fermionic Lagrangian. We then present the general constraints cast by these transformations and their products on the propagator of the simplest among coupled fermionic system, the one made with one fermion and its antifermion. Last, we put in correspondence the propagation of C eigenstates (Majorana fermions) and the criteria cast on their propagator by C and CP invariance.

Introduction

Fermions are usually treated, in most aspects of their phenomenology, as classical, though anticommuting, objects. Their Lagrangian is commonly endowed with a mass matrix though, for coupled systems,¹ this can only be a linear approximation in the vicinity of one among the physical poles of their full (matricial) propagator [1], [2]. In this perspective, the study of neutral kaons [1], and more specially of the role held, there, by discrete symmetries P, C, T and their products, has shown that subtle differences occur between the “classical” treatment obtained from a Lagrangian and a mass matrix, and the full quantum treatment dealing with their propagator. Using a classical approximation for fermions is a priori still more subject to caution since, in particular, their anticommutation is of quantum origin. This is why, after the work [1], we decided to perform a study of coupled fermionic systems in Quantum Field Theory, dealing especially with the propagator approach.² Treating fermions in a rigorous way is all the more important as the very nature of neutrinos, Dirac or Majorana, is still unknown, and that all theoretical results, concerning specially flavor mixing, have been mainly deduced from classical considerations.

The second and third parts of this work are dedicated to general statements concerning, first, symmetry transformations in general, then the discrete symmetries parity P, charge conjugation C, time-reversal T, and their products. It does not pretend to be original, but tries to make a coherent synthesis of results scattered in the literature. Starting from Wigner’s representation theorem [5] and Wightman’s point of view for symmetry transformations [6], we give the general rules of transformations of operators and of their hermitian conjugates by any unitary or antiunitary transformation. We then specialize to transforming Weyl spinors by P, C, T and their products, first when they are considered at the classical level (grassmanian wave functions), then at the quantum level (anticommuting operators).

The fourth part deals with the concept of invariance of a given theory. By taking the simple example of fermionic mass terms (Dirac and Majorana), we exhibit ambiguities and inconsistencies that arise in the transformations of a classical Lagrangian by antiunitary transformations. This motivates, like for neutral kaons [1], the propagator approach, which is the only safe way of deducing unambiguously the constraints cast by symmetry transformations on the Green functions of physical (propagating) particles, from which the S-matrix can be in principle reconstructed [6].

For the sake of simplicity, it is extensively investigated only in the case of the simplest among coupled fermionic systems, the one made with a single fermion and its antifermion; such a coupling, which concerns neutral particles, is indeed allowed by Lorentz invariance. This is the object of the fifth and last part of this work. We derive in full generality the constraints cast on the propagator by P, C, T, PC, PCT. We show that the physical (propagating) fermions can only be Majorana (C eigenstates) if their propagator satisfies the constraints cast by C or CP invariance.

The extension to several flavors, with its expected deeper insight into the issue of quantum mixing in connection with discrete symmetries, is currently under investigation.³

Section snippets

Generalities

In this paper, we shall note equivalently $ξ^{α} \overset{R}{\to} (ξ^{α})^{R} \equiv R \cdot ξ^{α}$ , where $ξ^{α}$ is a Weyl spinor (see Appendix A.1) and $R \cdot ξ^{α}$ its transformed by $R$ ; often the “ $\cdot$ ” will be omitted such that this transformed will also be noted $R ξ^{α}$ . The corresponding fermionic field operators will be put into square brackets, for example $[ξ^{α}], [ξ^{α}]^{R}$ , the latter being the transformed of the former by the transformation $R$ . Formally $[ξ^{α}]^{R} = (ξ^{α})^{R}$ .

The transition amplitude between two fermionic states is noted $〈 χ | ψ 〉$ ; this defines a

Parity transformation on grassmanian wave functions

We adopt the convention $P^{2} = - 1$ [12]. Then the transformation of spinors are $\begin{matrix} ξ^{α} (\vec{x}, t) \overset{P}{\to} i η_{\dot{α}} (- \vec{x}, t), η_{\dot{α}} (\vec{x}, t) \overset{P}{\to} i ξ^{α} (- \vec{x}, t), \\ ξ_{α} (\vec{x}, t) \overset{P}{\to} - i η^{\dot{α}} (- \vec{x}, t), η^{\dot{α}} (\vec{x}, t) \overset{P}{\to} - i ξ_{α} (- \vec{x}, t) . \end{matrix}$

The parity transformed of the complex conjugates are defined [12] as the complex conjugates of the parity transformed $P \cdot (ξ^{α})^{*} = (P \cdot ξ^{α})^{*};$ this ensures in particular that the constraints (13), (16) are satisfied. It yields $\begin{matrix} (ξ^{α})^{*} (\vec{x}, t) \overset{P}{\to} - i {(η_{\dot{α}})}^{*} (- \vec{x}, t), {(η_{\dot{α}})}^{*} (\vec{x}, t) \overset{P}{\to} - i (ξ^{α})^{*} (- \vec{x}, t), \\ (ξ_{α})^{*} (\vec{x}, t) \overset{P}{\to} i {(η^{\dot{α}})}^{*} (- \vec{x}, t), {(η^{\dot{α}})}^{*} (\vec{x}, t) \overset{P}{\to} i (ξ_{α})^{*} (- \vec{x}, t) . \end{matrix}$

For Dirac

Wightman’s point of view [6]

The invariance of a “theory” is expressed by the invariance of the vacuum and the invariance of all n-point functions; $O$ is then a product of fields at different space–time points and ( $\hat{O}$ being the transformed of $O$ ) $| 0 〉 = | \hat{0} 〉, 〈 0 | O | 0 〉 = 〈 0 | \hat{O} | 0 〉,$

∗ in the case of a unitary transformation $U$ , $〈 0 | O | 0 〉 \overset{sym}{=} 〈 0 | O^{U} | 0 〉 \overset{vacuuminv}{=} 〈 0^{U} | O^{U} | 0^{U} 〉, O^{U} = U^{- 1} OU;$ taking the example of parity and if $O = ϕ_{1} (x_{1}) ϕ_{2} (x_{2}) \dots ϕ_{n} (x_{n})$ , one has $O^{P} = P^{- 1} O P = ϕ_{1} (t_{1}, - {\vec{x}}_{1}) ϕ_{2} (t_{2}, - {\vec{x}}_{2}) \dots ϕ_{n} (t_{n}, - {\vec{x}}_{n})$ , such that parity invariance writes $〈 0 | ϕ_{1} (x_{1}) ϕ_{2} (x_{2}) \dots ϕ_{n} (x$

The fermionic propagator and discrete symmetries (1 fermion + its antifermion)

The fermionic propagator $Δ (x)$ is a matrix with a Lorentz tensorial structure, the matrix elements of which are the vacuum expectation values of $T$ -products of two fermionic operators: $T ψ (x) χ (y) = θ (x^{0} - y^{0}) ψ (x) χ (y) - θ (y^{0} - x^{0}) χ (y) ψ (x);$ the Lorentz indices of the two operators yield the tensorial structure of the matrix elements.

If, for example, one works in the fermionic basis $(ψ_{1}, ψ_{2}, ψ_{3}, ψ_{4})$ , and if $α, β \dots$ denote their Lorentz indices, the propagator is a $4 \times 4$ matrix $Δ (x)$ such that $Δ_{ij}^{α β} (x) = 〈ψ_{i}^{α} | Δ (x) | ψ_{j}^{β}〉 = 〈0 | T (ψ_{i}〉$

General conclusion

In this work, we have extended the propagator approach [3], [4], [1] to coupled fermionic systems. It is motivated, in particular, by the ambiguities that unavoidably occur when dealing with a classical fermionic Lagrangian endowed with a mass matrix. The goal of this formalism is, in particular, to determine at which condition the propagating neutral fermions, defined as the eigenstates, at the poles, of their full propagator, are Majorana. Due to the intricacies of this approach, we presently

Acknowledgments

Conversations, comments and critics with from V.A. Novikov, M.I. Vysotsky and J.B. Zuber are gratefully acknowledged.

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