Path Integral Measure Research Articles

Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman's Path-Integral Measure for Quantum Fields in the Presence of Gravity

Brownian Motion in the Hilbert Space of Quantum States along with the Ricci Flow and Stochastically Emergent Einstein-Hilbert Action: Formulating a Well-Defined Feynman's Path-Integral Measure for Quantum Fields in the Presence of Gravity

Gravitational edge mode in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 Jackiw-Teitelboim supergravity

We study the gravitational edge mode in the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 Jackiw-Teitelboim (JT) supergravity on the disk and its osp(2|1) BF formulation. We revisit the derivation of the finite-temperature Schwarzian action in the conformal gauge of the bosonic JT gravity through wiggling boundary and the frame fluctuation descriptions. Extending our method to N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 JT supergravity, we derive the finite-temperature super-Schwarzian action for the edge mode from both the wiggling boundary and the superframe field fluctuation. We emphasize the crucial role of the inversion of the super-Schwarzian derivative in elucidating the relation between the isometry and the OSp(2|1) gauging of the super-Schwarzian action. In osp(2|1) BF formulation, we discuss the asymptotic AdS condition. We employ the Iwasawa-like decomposition of OSp(2|1) group element to derive the super-Schwarzian action at finite temperature. We demonstrate that the OSp(2|1) gauging arises from inherent redundancy in the Iwasawa-like decomposition. We also discuss the path integral measure obtained from the Haar measure of OSp(2|1).

Brownian motion in the Hilbert space of quantum states and the stochastically emergent Lorentz symmetry: A fractal geometric approach from Wiener process to formulating Feynman’s path-integral measure for relativistic quantum fields

This paper aims to provide a consistent, finite-valued, and mathematically well-defined reformulation of Feynman’s path-integral measure for quantum fields obtained by studying the Wiener stochastic process in the infinite-dimensional Hilbert space of quantum states. This reformulation will undoubtedly have a crucial role in formulating quantum gravity within a mathematically well-defined framework. In fact, this study is fundamentally different from any previous research on the relationship between Feynman’s path-integral and the Wiener stochastic process. In this research, we focus on the fact that the classic Wiener measure is no longer applicable in infinite-dimensional Hilbert spaces due to fundamental differences between displacements in low and extremely high dimensions. Thus, an analytic norm motivated by the role of the fractal functions in the Wilsonian renormalization approach is worked out to properly characterize Brownian motion in the Hilbert space of quantum states on a compact flat manifold. This norm, the so-called fractal norm, pushes the rougher functions (physically the quantum states with higher energies) to the farther points of the Hilbert space until the fractal functions as the roughest ones are moved to infinity. Implementing the Wiener stochastic process with the fractal norm, results in a modified form of the Wiener measure called the Wiener fractal measure, which is a well-defined measure for Feynman’s path-integral formulation of quantum fields. Wiener fractal measure has a complicated formula of non-local terms but produces the Klein–Gordon action at the first order of approximation. Using complex integrals to compensate for the removal of non-local terms appearing in higher orders of approximation, the Wiener fractal measure turns into a complex measure and generates Feynman’s path-integral formulation of scalar quantum fields. This brings us to the main objective of this study. Finally, some various significant aspects of quantum field theory (such as renormalizability, RG flow, Wick rotation, regularization, etc.) are revisited by means of the analytical aspects of the Wiener fractal measure.

Vector-like symmetries and parity conservation in gauge theories with Yukawa couplings

Non-perturbative results in QCD-like theories can be derived employing positivity of the Euclidean path integral measure, as pioneered by Weingarten, Vafa, and Witten. We show that positivity of the measure can be generalized to parity-invariant theories with Yukawa couplings to fundamental scalars, provided the fermions are Dirac and carry a real representation of the gauge group. This result allows us to demonstrate the conservation of parity and vector-like flavor symmetries in such theories, as well as to derive exact inequalities among hadrons’ masses.

Neural network field theories: non-Gaussianity, actions, and locality

Both the path integral measure in field theory (FT) and ensembles of neural networks (NN) describe distributions over functions. When the central limit theorem can be applied in the infinite-width (infinite-N) limit, the ensemble of networks corresponds to a free FT. Although an expansion in 1/N corresponds to interactions in the FT, others, such as in a small breaking of the statistical independence of network parameters, can also lead to interacting theories. These other expansions can be advantageous over the 1/N -expansion, for example by improved behavior with respect to the universal approximation theorem. Given the connected correlators of a FT, one can systematically reconstruct the action order-by-order in the expansion parameter, using a new Feynman diagram prescription whose vertices are the connected correlators. This method is motivated by the Edgeworth expansion and allows one to derive actions for NN FT. Conversely, the correspondence allows one to engineer architectures realizing a given FT by representing action deformations as deformations of NN parameter densities. As an example, φ 4 theory is realized as an infinite-N NN FT.

Finiteness of quantum gravity with matter on a PL spacetime

We study the convergence of the path integral (PI) for general relativity with matter on a picewise linear (PL) spacetime that corresponds to a triangulation of a smooth manifold by using a PI measure that renders the pure gravity PI finite. This measure depends on a parameter p, and in the case when the matter content is just scalar fields, we show that the PI is absolutely convergent for 0,5 and not more than two scalar fields. In the case of Yang–Mills (YM) fields, we show that the PI is absolutely convergent for the U(1) group and 0,5. In the case of Dirac fermions, we show that the PI is absolutely convergent for any number of fermions and a sufficiently large p. When the matter content is given by scalars, YM fields and fermions, as in the case of the standard model (SM), we show that the PI is absolutely convergent for 52,5. Hence one can construct a finite quantum gravity theory on a PL spacetime such that the classical limit is general relativity coupled to the SM.

Lorentzian quantum gravity via Pachner moves: one-loop evaluation

Lorentzian quantum gravity is believed to cure the pathologies encountered in Euclidean quantum gravity, such as the conformal factor problem. We show that this is the case for the Lorentzian Regge path integral expanded around a flat background. We illustrate how a subset of local changes of the triangulation, so-called Pachner moves, allow to isolate the indefinite nature of the gravitational action at the discrete level. The latter can be accounted for by oppositely chosen deformed contours of integration. Moreover, we construct a discretization-invariant local path integral measure for 3D Lorentzian Regge calculus and point out obstructions in defining such a measure in 4D. We see the work presented here as a first step towards establishing the existence of the non-perturbative Lorentzian path integral for Regge calculus and related frameworks such as spin foams.An extensive appendix provides an overview of Lorentzian Regge calculus, using the recently established concept of the complexified Regge action, and derives useful geometric formulae and identities needed in the main text.

Note on the action with the Schwarzian at the stretched horizon

In this paper, we discuss the quantization of an interesting model of Carlip which appeared recently. It shows a way to associate boundary degrees of freedom to the stretched horizon of a stationary non-extremal black hole, as has been done in JT gravity for near-extremal black holes. The path integral now contains an integral over the boundary degrees of freedom, which are time reparametrizations of the stretched horizon keeping its length fixed. These boundary degrees of freedom can be viewed as elements of $Diff(S^1)/S^1$, which is the coadjoint orbit of an ordinary coadjoint vector under the action of the Virasoro group. From the symplectic form on this manifold, we obtain the measure in the boundary path integral. Doing a one-loop computation about the classical solution, we find that the one-loop answer is not finite, signalling that either the classical solution is unstable or there is an indefiniteness problem with this action, similar to the conformal mode problem in quantum gravity. Upon analytically continuing the field, the boundary partition function we get is independent of the inverse temperature and does not contribute to the thermodynamics at least at one-loop. This is in contrast to the study of near-extremal black holes in JT gravity, where the entire contribution to thermodynamics is from boundary degrees of freedom.

Anomalies from an effective field theory perspective

The path-integral measure of a gauge-invariant fermion theory is transformed under the chiral transformation and leads to an elegant derivation of the anomalous chiral Ward-Takahashi identities, as we know from the seminal work of Fujikawa. We present in this work an alternative and illuminating way to calculate the Jacobian in the path-integral measure from the covariant derivative expansion technique used in effective field theory. We present several ways to customize the crucial regularization such that the anomaly is located in the desired current, which is unprecedented within the path integral approach. We are then able to derive, in a transparent and unified way, the covariant, consistent, gravitational, and scale anomalies.

Axion anomalies

We study fermions derivatively coupled to axion-like or pseudoscalar fields, and show that the axial vector current of the fermions is not conserved in the limit where the fermion is massless. This apparent violation of the classical chiral symmetry is due to the background axion field. We compute the contributions to this anomalous Ward identity due to the pseudoscalar field alone, which arise in Minkowski space, as well as the effects due to an interaction with an external gravitational field. For the case of massless fermions, these interactions induce terms in the axion effective action that can be removed by the addition of local counterterms. We demonstrate that these counterterms are generated by the transformation of the path integral measure when transforming the theory from a form where the chiral symmetry is manifest to one where the symmetry is only apparent after using the classical equations of motion. We work perturbatively in Minkowski space and include the effects of interactions with a linearized gravitational field. Using the heat kernel method, we study the transformation properties of the path integral measure, and include the effects of non-linear gravity as well as interactions with gauge fields. Finally, we verify our relation by considering derivatively coupled fermions during pseudoscalar-driven inflation and computing the divergence of the axial current in de Sitter spacetime.

Effective Actions for Regge Piecewise Flat Quantum Gravity

We review the construction of the path integral and the corresponding effective action for the Regge formulation of General Relativity under the assumption that the short-distance structure of the spacetime is not a smooth 4-manifold, but a piecewise linear manifold based on a triangulation of a smooth 4-manifold. We point out that the exponentially damped 4-volume path-integral measure does not give a finite path integral, although it can be used for the construction of the perturbative effective action. We modify the 4-volume measure by multiplying it by an inverse power of the product of the edge-lengths such that the new measure gives a finite path integral while it retains all the nice features of the unmodified measure.

Path integral analysis of the axial anomaly in Very Special Relativity

In this note, we revise the problem of the axial anomaly in the framework of Very Special Relativity following Fujikawa’s path integral approach. We show nonperturbatively that no VSR contribution is present in the path integral measure in (3 + 1)-dimensional spacetime. Furthermore, we extend our results to (1 + 1) dimensions, as well as to the two-dimensional curved spacetime.

Wavefunction of the universe: Reparametrization invariance and field redefinitions of the minisuperspace path integral

We consider the Hartle–Hawking wavefunction of the universe defined as a Euclidean path integral that satisfies the “no-boundary proposal.” We focus on the simplest minisuperspace model that comprises a single scale factor degree of freedom and a positive cosmological constant. The model can be seen as a non-linear σ-model with a line-segment base. We reduce the path integral over the lapse function to an integral over the proper length of the base and use diffeomorphism-invariant measures for the ghosts and the scale factor. As a result, the gauge-fixed path integral is independent of the gauge. However, we point out that all field redefinitions of the scale factor degree of freedom yield different choices of gauge-invariant path-integral measures. For each prescription, we compute the wavefunction at the semi-classical level and find a different result. We resolve in each case the ambiguity in the form of the Wheeler–DeWitt equation at this level of approximation. By imposing that the Hamiltonians associated with these possibly distinct quantum theories are Hermitian, we determine the inner products of the corresponding Hilbert spaces and find that they lead to a universal norm, at least semi-classically. Quantum predictions are thus independent of the prescription at this level of approximation. Finally, all wavefunctions of the Hilbert spaces of the minisuperspace model we consider turn out to be non-normalizable, including the no-boundary states.

World quantum gravity: quantum test objects and Synge’s world function

A new path integral approach of quantum gravity based on relational variables and quantum test objects is presented. We take as a basic variables the squared invariant distance. This invariant quantity could be technically simpler to work with than variant quantities such as the metric tensor. It also could facilitate the studies of matter coupling and quantum spacetime causal structures. In contrast to approaches based on piecewise linear geometries, here gravity is captured by its effects on quantum test particles and fields. By an observation of Parker, under a Feynman sum a gravitational phase can be traded into a Van Vleck–Morette determinant term. This leads to a new candidate path integral for gravity, which in certain special cases can be computed efficient in the Lorentzian signature. We discuss some ambiguities left in the path integral measure, which invite further clarifications.

Axial Anomaly in SU(N) Yang-Mills Matrix Models.

The SU(N) Yang-Mills matrix model admits self-dual and anti-self-dual instantons. When coupled to N_{f} flavors of massless quarks, the Euclidean Dirac equation in an instanton background has n_{+} positive and n_{-} negative chirality zero modes. The vacua of the gauge theory are N-dimensional representations of SU(2), and the (anti-) self-dual instantons tunnel between two commuting representations, the initial one composed of r_{0}^{(1)} irreps and the final one with r_{0}^{(2)} irreps. We show that the index (n_{+}-n_{-}) in such a background is equal to a new instanton charge T_{new}=±[r_{0}^{(2)}-r_{0}^{(1)}]. Thus T_{new}=(n_{+}-n_{-}) is the matrix model version of the Atiyah-Singer index theorem. Further, we show that the path integral measure is not invariant under a chiral rotation, and relate the noninvariance of the measure to the index of the Dirac operator. Axial symmetry is broken anomalously, with the residual symmetry being a finite group. For N_{f} fundamental fermions, this residual symmetry is Z_{2N_{f}}, whereas for adjoint quarks it is Z_{4N_{f}}.

Cosmological constant and the use of cutoffs

Of the contributions to the cosmological constant, zero-point energy contributions scale as ${\ensuremath{\delta}}^{4}(0)\ensuremath{\sim}{\mathrm{\ensuremath{\Lambda}}}^{4}$ where $\mathrm{\ensuremath{\Lambda}}$ is an ultraviolet cutoff used to regulate the calculations. I show that such contributions vanish when calculated in perturbation theory. This demonstration uses a little-known modification to perturbation theory found by Honerkamp and Meetz and by Gerstein, Jackiw, Lee, and Weinberg which comes into play when using cutoffs and interactions with multiple derivatives, as found in chiral theories and gravity. In a path integral treatment, the new interaction arises from the path integral measure and cancels the ${\ensuremath{\delta}}^{4}(0)$ contributions. This reduces the sensitivity of the cosmological constant to the high energy cutoff, although it does not resolve the cosmological constant problem. The feature removes one of the common motivations for supersymmetry. It also calls into question some of the results of the asymptotic safety program. Covariance and quadratic cutoff dependence are also briefly discussed.

Covariant singularities in quantum field theory and quantum gravity

It is rather well-known that spacetime singularities are not covariant under field redefinitions. A manifestly covariant approach to singularities in classical gravity was proposed in [1]. In this paper, we start to extend this analysis to the quantum realm. We identify two types of covariant singularities in field space corresponding to geodesic incompleteness and ill-defined path integrals (hereby dubbed functional singularities). We argue that the former might not be harmful after all, whilst the latter makes all observables undefined. We show that the path-integral measure is regular in any four-dimensional theory of gravity without matter or in any theory in which gravity is either absent or treated semi-classically. This might suggest the absence of functional singularities in these cases, however it can only be confirmed with a thorough analysis, case by case, of the path integral. We provide a topological and model-independent classification of functional singularities using homotopy groups and we discuss examples of theories with and without such singularities.

Thermodynamic properties of the finite-temperature electron gas by the fermionic path integral Monte Carlo method

The recently developed quantum path integral Monte Carlo approach [Filinov et al., Phys. Rev. E 102(3), 033203 (2020)] has been applied for the entropy difference calculations for the strongly coupled degenerated uniform electron gas (UEG), a well-known model of simple metals. The used approach includes the Coulomb and exchange interaction of fermions in the basic Monte Carlo cell and its periodic images as well as the change of variables in the path integral measure and exchange determinant. Calculations have been carried out at finite temperature in the canonical ensemble. Presented results include the pair distribution functions, the isochors, and isotherms of pressure, internal energy, and the entropy change in the strongly coupled and degenerate UEG. The obtained data may be useful for density functional theory.

Towards a more algebraic footing for quantum field theory

The predictions of the standard model of particle physics are highly successful in spite of the fact that several parts of the underlying quantum field theoretical framework are analytically problematic. Indeed, it has long been suggested, by Einstein, Schrödinger and others, that analytic problems in the formulation of fundamental laws could be overcome by reformulating these laws without reliance on analytic methods namely, for example, algebraically. With this in mind, we focus here on the analytic ill-definedness of the quantum field theoretic Fourier and Legendre transforms of the generating series of Feynman graphs, including the path integral. To this end, we develop here purely algebraic and combinatorial formulations of the Fourier and Legendre transforms, employing rings of formal power series. These are all-purpose transform methods, i.e. their applicability is not restricted to quantum field theory. When applied in quantum field theory to the generating functionals of Feynman graphs, the new transforms are well-defined and thereby help explain the robustness and success of the predictions of perturbative quantum field theory in spite of analytic difficulties. Technically, we overcome here the problem of the possible divergence of the various generating series of Feynman graphs by constructing Fourier and Legendre transforms of formal power series that operate in a well-defined way on the coefficients of the power series irrespective of whether or not these series converge. Our new methods could, therefore, provide new algebraic and combinatorial perspectives on quantum field theoretic structures that are conventionally thought of as analytic in nature, such as the occurrence of anomalies from the path integral measure. In comparison, the use of formal power series in QFT by Bogolubov, Hepp, Parasiuk and Zimmermann concerned a different kind of divergencies, namely the UV divergencies of loop integrals and their renormalization.

World-Line Path Integral for the Propagator Expressed as an Ordinary Integral: Concept and Applications

The (Feynman) propagator $G(x_2,x_1)$ encodes the entire dynamics of a massive, free scalar field propagating in an arbitrary curved spacetime. The usual procedures for computing the propagator -- either as a time ordered correlator or from a partition function defined through a path integral -- requires introduction of a field $\phi(x)$ and its action functional $A[\phi(x)]$. An alternative, more geometrical, procedure is to define a propagator in terms of the world-line path integral which only uses curves, $x^i(s)$, defined on the manifold. I show how the world-line path integral can be reinterpreted as an ordinary integral by introducing the concept of effective number of quantum paths of a given length. Several manipulations of the world-line path integral become algebraically tractable in this approach. In particular, I derive an explicit expression for the propagator $G_{\rm QG}(x_2,x_1)$, which incorporates the quantum structure of spacetime through a zero-point-length, in terms of the standard propagator $G_{\rm std}(x_2,x_1)$, in an arbitrary curved spacetime. This approach also helps to clarify the interplay between the path integral amplitude and the path integral measure in determining the form of the propagator. This is illustrated with several explicit examples.