In recent work, Alexandre, Ellis, Millington and Seynaeve have extended the Goldstone theorem to non-Hermitian Hamiltonians that possess a discrete antilinear symmetry such as PT and possess a ...continuous global symmetry. They restricted their discussion to those realizations of antilinear symmetry in which all the energy eigenvalues of the Hamiltonian are real. Here, we extend the discussion to the two other realizations possible with antilinear symmetry, namely energies in complex conjugate pairs or Jordan-block Hamiltonians that are not diagonalizable at all. In particular, we show that under certain circumstances it is possible for the Goldstone boson mode itself to be one of the zero-norm states that are characteristic of Jordan-block Hamiltonians. While we discuss the same model as Alexandre, Ellis, Millington and Seynaeve, our treatment is quite different, though their main conclusion that one can have Goldstone bosons in the non-Hermitian case remains intact. We extend our analysis to a continuous local symmetry and find that the gauge boson acquires a nonzero mass by the Englert-Brout-Higgs mechanism in all realizations of the antilinear symmetry, except the one where the Goldstone boson itself has zero norm, in which case, and despite the fact that the continuous local symmetry has been spontaneously broken, the gauge boson remains massless.
In applications of the conformal gravity theory it has been shown that a scale of order 105 Mpc due to large scale inhomogeneities such as clusters of galaxies is imprinted on the rotation curves of ...galaxies. Here we show that this same scale is imprinted on recombination era anisotropies in the cosmic microwave background. We revisit an analysis due to Mannheim and Horne, to show that in the conformal gravity theory the particle horizon distance scale of metric signals that originate in the primordial nucleosynthesis era at 109∘ K can encompass the entire recombination era sky. Similarly, the particle horizon distance scale of acoustic signals that originate at 1013∘ K can also encompass the entire recombination era sky. We show that the amplitudes of metric fluctuations that originate in the nucleosynthesis era can grow by a factor of 1012 by recombination, and by a factor of 1018 by the current time. In addition we find that without any period of exponential expansion a fluctuation amplitude that begins at a temperature of order 1033∘ K can grow by a factor of 1060 by recombination and by a factor of 1066 by the current time.
We study cosmological perturbation theory in the cosmology associated with conformal gravity, establish the validity of the decomposition theorem for it, and then use the theorem to provide an exact ...solution to the theory in the recombination era. Central to our approach is the use of a fully gauge invariant formulation of the cosmological fluctuation equations. In the recombination era, not only is perturbation theory applicable, because of its specific structure in the conformal case, the fluctuation equations are found to greatly simplify. Using a master equation for scalar, vector, and tensor fluctuation modes, we show that the radial equations for the three-dimensional vector and tensor modes are, respectively, the same as those of scalar modes in five and seven spatial dimensions. This enables us to construct normalization conditions for the three-dimensional modes.
With fourth-order derivative theories leading to propagators of the generic ghostlike 1/(k2−M12)−1/(k2−M22) form, it would appear that such theories have negative norm ghost states and are not ...unitary. However on constructing the associated quantum Hilbert space for the free theory that would produce such a propagator, Bender and Mannheim found that the Hamiltonian of the free theory is not Hermitian but is instead PT symmetric, and that there are in fact no negative norm ghost states, with all Hilbert space norms being both positive and preserved in time. Even though perturbative radiative corrections cannot change the signature of a Hilbert space inner product, nonetheless it is not immediately apparent how such a ghostlike propagator would not then lead to negative probability contributions in loop diagrams. Here we obtain the relevant Feynman rules and show that all states obtained in cutting intermediate lines in loop diagrams have positive norm. Also we show that due to the specific way that unitarity (conservation of probability) is implemented in the theory, negative signatured discontinuities across cuts in loop diagrams are canceled by a novel and unanticipated contribution of the states in which tree approximation (no loop) graphs are calculated, an effect that is foreign to standard Hermitian theories. Perturbatively then, the fourth-order derivative theory with propagator 1/(k2−M12)−1/(k2−M22) is viable. Implications of our results for the pure massless 1/k4 propagator are also discussed.
Commutation or anticommutation relations quantized at equal instant time and commutation or anticommutation relations quantized at equal light-front time not only cannot be transformed into each ...other, they take completely different forms. While they would thus appear to describe different theories, we show that this is not in fact the case. By looking not at equal times but at unequal times, we show that unequal instant-time commutation or anticommutation relations are completely equivalent to unequal light-front time commutation or anticommutation relations. Light-front quantization and instant-time quantization are thus the same and thus describe the same theory, with it being only the restriction to equal times that makes them look different. However, for fermions there is a caveat, as the light-front anticommutation relations involve projection operators acting on the fermion fields. Nonetheless, not only can one still derive fermion unequal light-front time anticommutators starting from unequal instant-time ones, one can even derive unequal instant-time fermion anticommutators starting from unequal light-front time anticommutators even though the fermion projection operators that are relevant in the light-front case are not invertible. To establish the equivalence for gauge fields we present a quantization procedure that does not involve the zero-mode singularities that are commonly encountered in light-front gauge field studies. We also study time-ordered products of fields, and again show the equivalence despite the fact that there are additional terms in the fermion light-front case. We establish our results first for free theories, and then to all orders in interacting theories though comparison of the instant-time and light-front Lehmann representations. Finally, we compare instant-time Hamiltonians and light-front Hamiltonians and show that in the instant-time rest frame they give identical results.
A Hamiltonian H that is not Hermitian can still have a real and complete energy eigenspectrum if it instead is PT symmetric. For such Hamiltonians, three possible inner products have been considered ...in the literature, the V norm, the PT norm, and the C norm. Here, V is the operator that implements VHV−1=H†, the PT norm is the overlap of a state with its PT conjugate, and C is a discrete linear operator that always exists for any Hamiltonian that can be diagonalized. Here, we show that it is the V norm that is the most fundamental as it is always chosen by the theory itself. In addition, we show that the V norm is always equal to the PT norm if one defines the PT conjugate of a state to contain its intrinsic PT phase. We discuss the conditions under which the V norm coincides with the C operator norm and show that, in general, one should not use the linear C operator, but for the purposes that it is used one can instead use the antilinear PT operator itself.
We extend the CPT theorem to quantum field theories with non-Hermitian Hamiltonians and unstable states. Our derivation is a quite minimal one as it requires only the time-independent evolution of ...scalar products, invariance under complex Lorentz transformations, and a non-standard but nonetheless perfectly legitimate interpretation of charge conjugation as an antilinear operator. The first of these requirements does not force the Hamiltonian to be Hermitian. Rather, it forces its eigenvalues to either be real or to appear in complex conjugate pairs, forces the eigenvectors of such conjugate pairs to be conjugates of each other, and forces the Hamiltonian to admit of an antilinear symmetry. The latter two requirements then force this antilinear symmetry to be CPT, while forcing the Hamiltonian to be real rather than Hermitian. Our work justifies the use of the CPT theorem in establishing the equality of the lifetimes of unstable particles that are charge conjugates of each other. We show that the Euclidean time path integrals of a CPT-symmetric theory must always be real. In the quantum-mechanical limit the key results of the PT symmetry program of Bender and collaborators are recovered, with the C-operator of the PT symmetry program being identified with the linear component of the charge conjugation operator.
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