Abstract
In this chapter we propose to study the massive and massless theories of spin-1 and spin-2 fields through several approaches, each one of them providing a complementary viewpoint. As already mentioned in the introduction, the notions of degree of freedom and of dynamical field are not equivalent in non-local field theory. It is therefore important to first understand their equivalence in local field theory, and especially gauge theory, where not all fields propagate. We will thus see, in many different ways, how the field content splits into dynamical and non-dynamical fields and how this is related to the degrees of freedom of the theory.
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Notes
- 1.
Given the set of fields we consider, the spatial boundary conditions are zero at infinity.
- 2.
See Appendices A.3.2 and A.3.3 where we show this for \(\square ^{-1}_\mathrm{r}\) in real space and on arbitrary globally hyperbolic space-times. It can also be worked-out in Fourier space for both \(\square ^{-1}_\mathrm{r}\) and \(\square ^{-1}_\mathrm{F}\), since if the Fourier representation gives a finite result, i.e. if the operators are defined, then it is obvious that they commute with the derivatives and are also left-inverses.
- 3.
If the dynamical equations where of order n in the time-derivatives, this would give \(N_\mathrm{f} = n N_\mathrm{d}\).
- 4.
In realistic cases where \(j^{\mu }\) is also made of fundamental fields, the argument that the \(m = 0\) action is gauge-invariant because \(\partial _{\mu } j^{\mu } = 0\) no longer holds. Indeed, conservation equations can only hold for some field configurations, namely the on-shell ones, whereas a symmetry should hold for all field configurations in the action. There are then two possibilities. Either the \(A_{\mu } j^{\mu }\) term corresponds to non-minimal couplings to other fields through \(F_{\mu \nu }\), in which case it is itself gauge-invariant, or it emerges through minimal couplings that involve the covariant derivative \(\nabla \equiv \partial - i A\), in which case its variation is compensated by a non-trivial variation of A-independent terms. Then, because of that gauge symmetry, by Noether’s theorem for local symmetries we have that \(\partial _{\mu } j^{\mu } = 0\) on-shell. In the massive case, if the matter sector is unchanged, then we still have a global U(1) symmetry and it is thus Noether’s theorem for global symmetries which implies \(\partial _{\mu } j^{\mu } = 0\).
- 5.
It is actually the linearization of the Ricci scalar.
- 6.
This corresponds to the well-known fact that, when \(h_{\mu \nu }\) takes the form \(h_{\mu \nu } = \partial _{\mu } \partial _{\nu } \phi \) for some function \(\phi \), the Fierz–Pauli mass term is a total derivative. The generalization of this property to terms of cubic and higher order in \(\partial _{\mu } \partial _{\nu } \phi \) gives rise to the Galileon family of operators [3].
- 7.
Weak equality “\(\approx \)” holds for “\(=\) up to the addition of terms that are zero on the constrained hypersurface”.
- 8.
A constraint is “first class” if its Poisson bracket with any other constraint and the Hamiltonian is weakly zero. A constraint that is not first class is called “second class”.
- 9.
See for instance [9] for details on this formalism.
- 10.
Note that the second term here is needed because \(\mathcal{O}\) can depend on the source which has its own time-dependence. The \(\partial _t\) operator will of course not act on the smearing fields \(f_i\), \(g_i\) and \(A_0\).
- 11.
- 12.
This will become clear in the next section where we will deduce the transformations of the harmonic variables under the gauge symmetry.
- 13.
Note that the square-root is real because \(\Delta \big ( \Delta - m^2 \big )^{-1}\) is positive-definite as it can be seen by using its Fourier representation.
- 14.
After deriving this result we were informed by S. Deser (private communication) that a similar form was obtained in an old and little known paper [11]. Interestingly enough, this paper appeared in 1966, that is 14 years before the introduction of gauge-invariant variables by Bardeen [10] in cosmological perturbation theory.
- 15.
Since the mapping between \(\Omega \) and \(\Phi _0\) is singular as \(M \rightarrow 0\), we should actually check this result by working directly on the \(M = 0\) point, in which case integrating out \(\lambda \) to fix \(\psi '\) gives \(S_\mathrm{scal.} = 0\).
- 16.
The Ward identity is usually presented as a direct consequence of gauge symmetry and it can therefore appear as a surprise that it still holds in the massive case. However, note that one can also derive the identity by simply using the operator equation \(\partial _{\mu } \hat{A}^{\mu } = 0\), which is valid in the massive case, when computing correlation functions with on-shell external momenta \(\partial _{\mu } \langle 0 | T\left\{ \hat{A}_{\mu }(x) \ldots \right\} | 0 \rangle = 0\). Thus, the Ward identity still holds in massive electrodynamics, not because \(\partial _{\mu } A^{\mu }\) contains no propagating degrees of freedom as in the massless case, but because \(\partial _{\mu } A^{\mu }\) is simply zero on-shell.
- 17.
For non-linear theories the gauge-fixing term breaks the unitarity of the S-matrix and one must also include Faddeev–Popov fields to restore it.
- 18.
Demanding heavier sources \(m_\mathrm{s} > m\) and no loops implies that the virtual photon can never be on-shell, i.e. it is never a “real” photon. Alternatively, if \(m_\mathrm{s} < m\), then the process in which the photon is on-shell would be kinematically allowed, in which case the propagator would be singular, implying an infinite probability for this process to occur. As in the case of any heavy particle, the resolution of this apparent problem comes by noting that the heavy particle becomes unstable precisely when \(m_\mathrm{s} < m\), since it can then disintegrate into the source’s particles. By the optical theorem, we then have that the imaginary part of the vacuum polarization diagram becomes non-zero. Since that diagram is responsible for shifting the mass m under radiative corrections in the propagator, we get that the poles of the renormalized propagator have a non-vanishing imaginary part. Thus, the case \(k^2 = -m_\mathrm{ren}^2\), where \(m_\mathrm{ren}\) is the renormalized mass of the photon, is not a singularity of the renormalized propagator but rather the maximum of the so-called “Breit–Wigner” resonance.
- 19.
The original action is then obtained by integrating-out \(\psi \).
- 20.
This is why any other kinetic term than \(F_{\mu \nu }F^{\mu \nu }\) for a vector field implies a ghost by the way.
- 21.
This is not a surprise since the right-hand side of (2.7.2) is transverse.
- 22.
This is actually not really possible for the time coordinate since \(A_{\mu }\) will in general not vanish at future infinity because of the waves generated by the source. One should rather use a Laplace transform for t since the support of \(A_{\mu }\) is bounded in the past.
- 23.
Note that in [17, 18] the authors erroneously concluded that this theory propagates only the d-tensor part of \(h_{\mu \nu }\), i.e. it has the same dynamical content as the massless theory, because it has the same tensor structure (adding a gauge-fixing term and inverting one finds that the saturated propagator is indeed (2.5.18)). Their argument is that one has precisely integrated-out the Stückelbergs which correspond to the d-vector and d-scalar modes, so that the latter do not appear in this equation. As we have seen, this is not true because the Stückelbergs do not represent the dynamical fields that are activated by the mass. Moreover, it is not the tensor structure of the propagator alone which determines the dynamical content, otherwise the latter would be the same in massless and massive electrodynamics. As we have also seen, the presence of the mass is important, because it will affect the conservation equation of the source in Fourier space. Indeed, as we pointed out in [1], by expressing the saturated propagator (2.5.18) in terms of the harmonic variables of the conserved sources, we get (2.5.20) with M having both a positive and a negative eigenvalue (the ghost pole). We then have that \(M \rightarrow 0\) as \(m_\mathrm{s} \rightarrow m \rightarrow 0\) so that we have no vDVZ discontinuity, as expected. However, for \(m \ne 0\), all the independent components of the source are present and thus so are all the dynamical fields of the local theory.
- 24.
Up to Klein–Gordon operators.
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Mitsou, E. (2016). Linear Massless/Massive Gauge Theories. In: Infrared Non-local Modifications of General Relativity . Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-31729-8_2
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