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Source PDF: mmathphys.physics.ox.ac.uk/sites/g/files/vzsiac2991/files/mmathphys/documents/media/a15088h1.pdf

The solutions use natural units $\hslash =c=1$ and the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$. In question 1, ${\gamma }^5={\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ is exactly the paper's convention: it contains no factor $i$. The qualifications in iii, d and d distinguish the printed expressions from the results of charge conservation, identical final-state symmetry factors and fermionic time ordering.

Put $z=x-y$ and $d{\Pi }_p=d^3\mathbf{p}/[(2\pi {)}^{3}2E_{\mathbf{p}}]$. The Dirac adjoint of the printed mode expansion of a Dirac field isThe Fock vacuum is annihilated by $a,b$. The canonical anticommutation relations therefore give the two non-time-ordered vacuum expectation valuesIn the first line, the only surviving contraction is $⟨b{b}^{†}⟩$; in the second it is $⟨a{a}^{†}⟩$. Both of these unordered contractions have positive coefficients.

However, fermionic time ordering includes a minus sign when it interchanges two odd fields:Thus the result following from the stated canonical anticommutation relations and standard time ordering isThe second term printed with a plus sign is not correct for standard fermionic time ordering. Its plus-plus expression would be an ungraded chronological product. The missing minus cannot be absorbed by a phase choice for $u$ or $v$, since each spin sum contains a spinor and its adjoint.

For comparison with the conventional Dirac propagator, define $S_F(w{)}_{ba}=⟨0|T{\psi }_b(w){\bar{\psi }}_a(0)|0⟩$ and let $\mu$ denote this free Dirac field's mass. ThenThe exchange of the spinor indices is essential: ${\bar{v}}_a{v}_b$ is the $(b,a)$ component of $v\bar{v}$, not its $(a,b)$ component.

The corrected expression in d is a vacuum two-point function, or Wick contraction, of the free Dirac field. It equals the negative spinor transpose of the usual Dirac propagator with reversed spacetime arguments. It supplies the internal fermion lines in Feynman diagrams, rather than a position-space detection probability.

For $x^0>y^0$, $\psi (y)$ creates an antiparticle and $\bar{\psi }(x)$ annihilates it. The $v\bar{v}$ term describes positive-energy antiparticle propagation from $y$ to $x$. For $y^0>x^0$, $\bar{\psi }(x)$ creates a particle and $\psi (y)$ annihilates it. The $u\bar{u}$ term describes positive-energy particle propagation from $x$ to $y$, with the minus sign required by exchanging the odd operators in fermionic time ordering. Neither term describes a physical negative-energy excitation.

The Heaviside step functions select these chronological alternatives. The factors $e^{\mp ip\cdot z}$ are on-shell plane waves with $p^0=E_{\mathbf{p}}>0$, and $d{\Pi }_p$ is the positive-energy Lorentz-invariant phase-space measure. The sum over $s$ includes the two spin states. With standard relativistic spinor normalization, the fermion spin sums areThey encode the spinor structure and solve the corresponding on-shell Dirac equations. In particular, the corrected answer can also be writtenThe Feynman i-epsilon prescription specifies the temporal boundary conditions of the Green function; its positive- and negative-frequency residues reproduce the two terms and their relative minus sign. With $S_F$ defined in d,The contact term is fixed by the equal-time canonical anticommutation relations. Choosing a value for $\theta (0)$ does not change the unequal-time sign demonstrated above.

Finally, microcausality requires field anticommutators to vanish at spacelike separation; the time-ordered vacuum two-point function itself need not vanish there. These facts are compatible with relativistic causality. The plus-plus expression printed in the paper lacks the fermionic sign needed for this standard Dirac propagator interpretation.

A two-field vacuum expectation value. Its ordered, time-ordered and anticommutator versions are different distributions. For a free Dirac field, fermionic time ordering combines the positive-energy particle and antiparticle contributions into a Dirac propagator. Its nonzero value at spacelike separation does not violate microcausality, which constrains the field anticommutator.