In the context of field theories with complex poles, we scrutinize four inequivalent ways of defining the scattering amplitudes, each forfeiting one or more tenets of standard quantum field theory while preserving the others: (i) a textbook Wick rotation by analytic continuation of the external momenta from Euclidean to Lorentzian signature (no optical theorem), (ii) the Lee-Wick-Nakanishi prescription, integrating along a certain contour in the complex energy plane (no Lorentz invariance), (iiii) the fakeon prescription, where, in addition, spatial momenta are integrated on a complex path defined by the locus of singularities of the loop integrand (no analyticity of the amplitude) and (iv) to work directly on Minkowski spacetime, which violates the optical theorem and also bars power-counting renormalizability. In general, mixed Euclidean-Lorentzian prescriptions for internal and external momenta in loop integrals break Lorentz invariance, regardless of the type of masses involved. We conclude that, of the above four options, only the fakeon prescription is physically viable and can have applications to quantum gravity.

Italy

Damiano Anselmi

Fabio Briscese

Spain

Gianluca Calcagni

Leonardo Modesto

Amplitude prescriptions in field theories with complex poles

We study gauge theories and quantum gravity diagrammatically in a finite interval of time <math><mi>τ</mi></math>, on a compact space manifold <math><mi>Ω</mi></math>. The initial, final, and boundary conditions are formulated in gauge invariant and general covariant ways by means of purely virtual extensions of the theories, which allow us to “trivialize” the local symmetries and switch to invariant fields (the invariant metric tensor, invariant quark, and gluon fields, etc.). The evolution operator <math><mi>U</mi><mo>(</mo><msub><mi>t</mi><mi>f</mi></msub><mo>,</mo><msub><mi>t</mi><mi>i</mi></msub><mo>)</mo></math> is worked out for arbitrary initial and final states, as well as general boundary conditions on <math><mo>∂</mo><mi>Ω</mi></math>. We show that <math><mi>U</mi><mo>(</mo><msub><mi>t</mi><mi>f</mi></msub><mo>,</mo><msub><mi>t</mi><mi>i</mi></msub><mo>)</mo></math> is well defined and diagrammatically unitary for every <math><mi>τ</mi><mo>=</mo><msub><mi>t</mi><mi>f</mi></msub><mo>−</mo><msub><mi>t</mi><mi>i</mi></msub><mo>&lt;</mo><mi>∞</mi></math>. The formulation is extended to include purely virtual particles. In quantum gravity, where the cosmological constant <math><msub><mi>Λ</mi><mi>C</mi></msub></math> challenges the definition of an <math><mi>S</mi></math> matrix, the results allow us to prove unitarity at <math><mi>τ</mi><mo>&lt;</mo><mi>∞</mi></math>. We work out the frequencies and eigenfunctions in some explicit examples, including Yang-Mills theory on the finite cylinder.

Gauge theories and quantum gravity in a finite interval of time on a compact space manifold

Quantum gravity is extended to include purely virtual “cloud sectors”, which allow us to define a complete set of point-dependent observables, including a gauge invariant metric and gauge invariant matter fields, and calculate their off-shell correlation functions perturbatively. The ordinary on-shell correlation functions and the S matrix elements are unaffected. Each extra sector is made of a cloud field, its anticommuting partner, a “cloud-fixing” function and a cloud Faddeev-Popov determinant. The additional fields are purely virtual, to ensure that no ghosts propagate. The extension is unitary. In particular, the off-shell, diagrammatic version of the optical theorem holds. The one-loop two-point functions of dressed scalars, vectors and gravitons are calculated. Their absorptive parts are positive, cloud independent and gauge independent, while they are unphysical if non purely virtual clouds are used. We illustrate the differences between our approach to the problem of finding a complete set of observables in quantum gravity and other approaches available in the literature.

Purely virtual extension of quantum field theory for gauge invariant fields: quantum gravity

We provide a diagrammatic formulation of perturbative quantum field theory in a finite interval of time τ, on a compact space manifold Ω. We explain how to compute the evolution operator U(t f , t i) between the initial time t i and the final time t f = t i + τ, study unitarity and renormalizability, and show how to include purely virtual particles, by rendering some physical particles (and all the ghosts, if present) purely virtual. The details about the restriction to finite τ and compact Ω are moved away from the internal sectors of the diagrams (apart from the discretization of the three-momenta), and coded into external sources. Unitarity is studied by means of the spectral optical identities, and the diagrammatic version of the identity U †(t f , t i)U(t f , t i) = 1. The dimensional regularization is extended to finite τ and compact Ω, and used to prove, under general assumptions, that renormalizability holds whenever it holds at τ = ∞, Ω = ℝ3. Purely virtual particles are introduced by removing the on-shell contributions of some physical particles, and the ghosts, from the core diagrams, and trivializing their initial and final conditions. The resulting evolution operator U ph(t f , t i) is unitary, but does not satisfy the more general identity U ph(t 3 , t 2)U ph(t 2 , t 1) = U ph(t 3 , t 1). As a consequence, U ph(t f , t i) cannot be derived from a Hamiltonian in a standard way, in the presence of purely virtual particles.

Quantum field theory of physical and purely virtual particles in a finite interval of time on a compact space manifold: diagrams, amplitudes and unitarity

We study the free and dressed propagators of physical and purely virtual particles in a finite interval of time τ and on a compact space manifold Ω, using coherent states. In the free-field limit, the propagators are described by the entire function (e  z  – 1 – z)/z 2, whose shape on the real axis is similar to the one of a Breit-Wigner function, with an effective width around 1/τ. The real part is positive, in agreement with unitarity, and remains so after including the radiative corrections, which shift the function into the physical half plane. We investigate the effects of the restriction to finite τ on the problem of unstable particles vs resonances, and show that the muon observation emerges from the right physical process, differently from what happens at τ = ∞. We also study the case of purely virtual particles, and show that, if τ is small enough, there exists a situation where the geometric series of the self-energies is always convergent. The plots of the dressed propagators show testable differences: while physical particles are characterized by the usual, single peak, purely virtual particles are characterized by twin peaks.

Propagators and widths of physical and purely virtual particles in a finite interval of time

We extend quantum field theory by including purely virtual “cloud” sectors, to define physical off-shell correlation functions of gauge invariant quark and gluon fields, without affecting the S matrix amplitudes. The extension is made of certain cloud bosons, plus their anticommuting partners. Both are quantized as purely virtual, to ensure that they do not propagate ghosts. The extended theory is renormalizable and unitary. In particular, the off-shell, diagrammatic version of the optical theorem holds. We calculate the one-loop two-point functions of dressed quarks and gluons, and show that their absorptive parts are gauge independent, cloud independent and positive (while they are generically unphysical if the cloud sectors are not purely virtual). A gauge/cloud duality simplifies the computations and shows that the gauge choice is just a particular cloud. It is possible to dress every field insertion with a different cloud. We compare the purely virtual extension to previous approaches to similar problems.

Purely virtual extension of quantum field theory for gauge invariant fields: Yang–Mills theory

We formulate a new quantization principle for perturbative quantum field theory, based on a minimally non time-ordered product, and show that it gives the theories of physical particles and purely virtual particles. Given a classical Lagrangian, the quantization proceeds as usual, guided by the time-ordered product, up to the common scattering matrix S, which satisfies a unitarity or a pseudounitarity equation. The physical scattering matrix S ph is built from S, by gluing S diagrams together into new diagrams, through non time-ordered propagators. We classify the most general way to gain unitarity by means of such operations, and point out that a special solution “minimizes” the time-ordering violation. We show that the scattering matrix S ph given by this solution coincides with the one obtained by turning the would-be ghosts (and possibly some would-be physical particles) into purely virtual particles (fakeons). We study tricks to descend and ascend in a unique way among diagrams, and illustrate them in several examples: the ascending chain from the bubble to the hexagon, at one loop; the box with diagonal, at two loops; other diagrams, with more loops.

A new quantization principle from a minimally non time-ordered product

We reconsider the Lee-Wick (LW) models and compare their properties to the properties of the models that contain purely virtual particles. We argue against the LW premise that unstable particles can be removed from the sets of incoming and outgoing states in scattering processes. The removal leads to a non-Hermitian classical limit, besides clashing with the observation of the muon. If, on the other hand, all the states are included, the LW models have a Hamiltonian unbounded from below or negative norms. Purely virtual particles, on the contrary, lead to a Hermitian classical limit and are absent from the sets of incoming and outgoing states without implications on the observation of long-lived unstable particles. We give a vademecum to summarize the properties of most options to treat abnormal particles. We study a method to remove the LW ghosts only partially, by saving the physical particles they contain. Specifically, we replace a LW ghost with a certain superposition of a purely virtual particle and an ordinary particle, and drop only the former from the sets of the external states. The trick can be used to make the Pauli-Villars fields consistent and observable, without sending their masses to infinity, or to build a finite QED, by tweaking the original Lee-Wick construction. However, it has issues with general covariance, so it cannot be applied as is to quantum gravity, where a manifestly covariant decomposition requires the introduction of a massive spin-2 multiplet.

Purely virtual particles versus Lee-Wick ghosts: Physical Pauli-Villars fields, finite QED, and quantum gravity

We study the resummation of self-energy diagrams into dressed propagators in the case of purely virtual particles and compare the results with those obtained for physical particles and ghosts. The three geometric series differ by infinitely many contact terms, which do not admit well-defined sums. The peak region, which is outside the convergence domain, can only be reached in the case of physical particles, thanks to analyticity. In the other cases, nonperturbative effects become important. To clarify the matter, we introduce the energy resolution ∆E around the peak and argue that a “peak uncertainty” ∆E ≳ ∆E min ≃ Γf/2 around energies E ≃ m f expresses the impossibility to approach the fakeon too closely, m f being the fakeon mass and Γf being the fakeon width. The introduction of ∆E is also crucial to explain the observation of unstable long-lived particles, like the muon. Indeed, by the common energy-time uncertainty relation, such particles are also affected by ill-defined sums at ∆E = 0, whenever we separate their observation from the observation of their decay products. We study the regime of large Γf, which applies to collider physics (and situations like the one of the Z boson), and the regime of small Γf, which applies to quantum gravity (and situations like the one of the muon).

Dressed propagators, fakeon self-energy and peak uncertainty

We prove spectral optical identities in quantum field theories of physical particles (defined by the Feynman iϵ prescription) and purely virtual particles (defined by the fakeon prescription). The identities are derived by means of purely algebraic operations and hold for every (multi)threshold separately and for arbitrary frequencies. Their major significance is that they offer a deeper understanding on the problem of unitarity in quantum field theory. In particular, they apply to “skeleton” diagrams, before integrating on the space components of the loop momenta and the phase spaces. In turn, the skeleton diagrams obey a spectral optical theorem, which gives the usual optical theorem for amplitudes, once the integrals on the space components of the loop momenta and the phase spaces are restored. The fakeon prescription/projection is implemented by dropping the thresholds that involve fakeon frequencies. We give examples at one loop (bubble, triangle, box, pentagon and hexagon), two loops (triangle with “diagonal”, box with diagonal) and arbitrarily many loops. We also derive formulas for the loop integrals with fakeons and relate them to the known formulas for the loop integrals with physical particles.