The reality and dimension of space and the complexity of quantum mechanics

The dimension (and signature) of space is a result of distances being real numbers and quantum mechanical state functions being complex ones; it is an inescapable consequence of quantum mechanics and group theory. So nonrelativistic quantum mechanics cannot be complete (it requiresad hoc additional assumptions) and consistent (nor can classical physics), leading to relativity, quantum mechanics, and field theory. Implications of the constraints of consistency and physical reasonableness and of group theory for the structure of these theories are considered. It appears that there are simple, perhaps unavoidable reasons for the laws of physics, the nature of the world they describe, and the space in which they act.     

Axiomatic Quantum Field Theory in Curved Spacetime

The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincaré invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globally hyperbolic curved spacetimes, it is essential that the theory be formulated in an entirely local and covariant manner, without assuming the presence of a preferred state. We propose a new framework for quantum field theory, in which the existence of an Operator Product Expansion (OPE) is elevated to a fundamental status, and, in essence, all of the properties of the quantum field theory are determined by its OPE. We provide general axioms for the OPE coefficients of a quantum field theory. These include a local and covariance assumption (implying that the quantum field theory is constructed in a local and covariant manner from the spacetime metric and other background structure, such as time and space orientations), a microlocal spectrum condition, an “associativity” condition, and the requirement that the coefficient of the identity in the OPE of the product of a field with its adjoint have positive scaling degree. We prove curved spacetime versions of the spin-statistics theorem and the PCT theorem. Some potentially significant further implications of our new viewpoint on quantum field theory are discussed.     

The Theory of (Exclusively) Local Beables

It is shown how, starting with the de Broglie–Bohm pilot-wave theory, one can construct a new theory of the sort envisioned by several of QM’s founders: a Theory of Exclusively Local Beables (TELB). In particular, the usual quantum mechanical wave function (a function on a high-dimensional configuration space) is not among the beables posited by the new theory. Instead, each particle has an associated “pilot-wave” field (living in physical space). A number of additional fields (also fields on physical space) maintain what is described, in ordinary quantum theory, as “entanglement.” The theory allows some interesting new perspective on the kind of causation involved in pilot-wave theories in general. And it provides also a concrete example of an empirically viable quantum theory in whose formulation the wave function (on configuration space) does not appear—i.e., it is a theory according to which nothing corresponding to the configuration space wave function need actually exist. That is the theory’s raison d’etre and perhaps its only virtue. Its vices include the fact that it only reproduces the empirical predictions of the ordinary pilot-wave theory (equivalent, of course, to the predictions of ordinary quantum theory) for spinless non-relativistic particles, and only then for wave functions that are everywhere analytic. The goal is thus not to recommend the TELB proposed here as a replacement for ordinary pilot-wave theory (or ordinary quantum theory), but is rather to illustrate (with a crude first stab) that it might be possible to construct a plausible, empirically viable TELB, and to recommend this as an interesting and perhaps-fruitful program for future research.     

Limits of applicability of Gupta's theory

The applicability of the quantum theory of a weak gravitational field (Gupta's theory) is examined on the basis of the quasiclassical generally relativistic point of view. It is shown that in standard quantum theory the energy of a particle cannot be arbitrary but is bounded both below and above. These bounds arise because it is impossible to treat the region of interaction of elementary particles as a part of flat space. The lower limit depends on the curvature tensor of the external gravitational field, while the upper is determined by the gravitational field of the particle itself.     

Scalar quantum field theory with a complex cubic interaction

In this Letter it is shown that an i phi(3) quantum field theory is a physically acceptable model because the spectrum is positive and the theory is unitary. The demonstration rests on the perturbative construction of a linear operator C, which is needed to define the Hilbert space inner product. The C operator is a new, time-independent observable in PT-symmetric quantum field theory.     

Field theories on a lattice

Lattice field theories are described as a way to regularize continuum quantum field theories. They are obtained by replacing ordinary space time by a lattice, space time derivatives by suitable differences and Minkowski by Euclidean space. The connection between a quantum field theory isd space dimension and classical statistical mechanics in (d+1) dimensions is brought outvia elementary examples. The problem of regaining the continuum limit and of handling nonabelian gauge theories are briefly discussed.     

Quantum Field Theory with a Minimal Length Induced from Noncommutative Space

From the inspection of noncommutative quantum mechanics, we obtain an approximate equivalent relation for the energy dependence of the Planck constant in the noncommutative space, which means a minimal length of the space. We find that this relation is reasonable and it can inherit the main properties of the noncommutative space. Based on this relation, we derive the modified Klein-Gordon equation and Dirac equation. We investigate the scalar field and φ4 model and then quantum electrodynamics in our theory, and derive the corresponding Feynman rules. These results may be considered as reasonable approximations to those of noncommutative quantum field theory. Our theory also shows a connection between the space with a minimal length and the noncommutative space.     

Lifting a conformal field theory from d-dimensional flat space to (d+1)-dimensional AdS space

A quantum field theory on anti-de Sitter space can be constructed from a conformal field theory on its boundary Minkowski space by an inversion of the holographic mapping. The resulting theory is defined by its Green functions and is conformally covariant. The structure of operator product expansions is carried over to AdS space. We show that this method yields a higher spin field theory HS(4) from the minimal conformal O(N) sigma model in three dimensions.     

Einstein–Cartan gravity,Asymptotic Safety,and the running Immirzi parameter

In this paper we analyze the functional renormalization group flow of quantum gravity on the Einstein–Cartan theory space. The latter consists of all action functionals depending on the spin connection and the vielbein field (co-frame) which are invariant under both spacetime diffeomorphisms and local frame rotations. In the first part of the paper we develop a general methodology and corresponding calculational tools which can be used to analyze the flow equation for the pertinent effective average action for any truncation of this theory space. In the second part we apply it to a specific three-dimensional truncated theory space which is parametrized by Newton’s constant, the cosmological constant, and the Immirzi parameter. A comprehensive analysis of their scale dependences is performed, and the possibility of defining an asymptotically safe theory on this hitherto unexplored theory space is investigated. In principle Asymptotic Safety of metric gravity (at least at the level of the effective average action) is neither necessary nor sufficient for Asymptotic Safety on the Einstein–Cartan theory space which might accommodate different “universality classes” of microscopic quantum gravity theories. Nevertheless, we do find evidence for the existence of at least one non-Gaussian renormalization group fixed point which seems suitable for the Asymptotic Safety construction in a setting where the spin connection and the vielbein are the fundamental field variables.     

On the role of coherent states in quantum foundations

Coherent states, and the Hilbert space representations they generate, provide ideal tools to discuss classical/quantum relationships. In this paper we analyze three separate classical/quantum problems using coherent states, and show that useful connections arise among them. The topics discussed are: (1) a truly natural formulation of phase space path integrals; (2) how this analysis implies that the usual classical formalism is “simply a subset” of the quantum formalism, and thus demonstrates a universal coexistence of both the classical and quantum formalisms; and (3) how these two insights lead to a complete analytic solution of a formerly insoluble family of nonlinear quantum field theory models.     

One-loop approximation of Møller scattering in generalized Krein-space quantization

It has been shown that the negative-norm states necessarily appear in a covariant quantization of the free minimally coupled scalar field in de Sitter spacetime. In this processes ultraviolet and infrared divergences have been automatically eliminated. A natural renormalization of the one-loop interacting quantum field in Minkowski spacetime (λφ ⁴) has been achieved through the consideration of the negative-norm states defined in Krein space. It has been shown that the combination of quantum field theory in Krein space together with consideration of quantum metric fluctuation, results in quantum field theory without any divergences. Pursuing this approach, we express Wick’s theorem and calculate Møller scattering in the one-loop approximation in generalized Krein space. The mathematical consequence of this method is the disappearance of the ultraviolet divergence in the one-loop approximation.     

On the meaning of a non-renormalizable theory of gravitation

The renormalizability of quantum gravity remains an open question while it has been established recently that quantum gravity in the presence of standard sources is non-renormalizable. In view of traditional confusion and ambiguities surrounding non-renormalizable quantum field theories, it has been felt that physical theories must be renormalizable. Recently a new, nonperturbative view of non-renormalizable theories has been suggested that may have relevance for various interactions including gravity and various sources. In a path integral approach to quantum field theory such a view attributes ‘hard cores’ in the space of field histories to non-renormalizable interactions. Just as with more familiar ‘hard cores’, turning off the interaction does not completely remove all effects of the potential. Consequently the interacting theory is not even continuously connected to the usual free theory, but rather to an alternative ‘pseudo-free’ theory that incorporates the vestiges of the ‘hard cores’. Some insight into what is the significance and interpretation of non-renormalizable interactions can be gleaned from exactly soluble models. Application of this philosophy of non-renormalizable interactions is discussed for the gravitational field in interaction with some standard sources.     

A Relativistic Gauge Theory of Nonlinear Quantum Mechanics and Newtonian Gravity

A new kind of gauge theory is introduced, where the minimal coupling and corresponding covariant derivatives are defined in the space of functions pertaining to the functional Schrödinger picture of a given field theory. While, for simplicity, we study the example of a \(\mathcal{U}(1)\) symmetry, this kind of gauge theory can accommodate other symmetries as well. We consider the resulting relativistic nonlinear extension of quantum mechanics and show that it incorporates gravity in the (0+1)-dimensional limit, similar to recently studied Schrödinger-Newton equations. Gravity is encoded here into a universal nonlinear extension of quantum theory. A probabilistic interpretation (Born’s rule) holds, provided the underlying model is scale free.     

Relativistic Gamow Vectors

Whereas in Dirac quantum mechanics and relativistic quantum field theory one uses Schwartz space distributions, the extensions of the Hilbert space that we propose uses Hardy spaces. The in- and out-Lippmann-Schwinger kets of scattering theory are functionals in two rigged Hilbert space extensions of the same Hilbert space. This hypothesis also allows to introduce generalized vectors corresponding to unstable states, the Gamow kets. Here the relativistic formulation of the theory of unstable states is presented. It is shown that the relativistic Gamow vectors of the unstable states, defined by a resonance pole of the S-matrix, are classified according to the irreducible representations of the semigroup of the Poincaré transformations (into the forward light cone). As an application the problem of the mass definition of the intermediate vector boson Z is discussed and it is argued that only one mass definition leads to the exponential decay law, and that is not the standard definition of the on-the-mass-shell renormalization scheme.     

Quantum mechanics and the physical reality concept

The difference between the measurement bases of classical and quantum mechanics is often interpreted as a loss of reality arising in quantum mechanics. In this paper it is shown that this apparent loss occurs only if one believes that refined everyday experience determines the Euclidean space as the real space, instead of considering this space, both in classical and quantum mechanics, as a theoretical construction needed for measurement and representing one part of a dualistic space conception. From this point of view, Einstein's program of a unified field theory can be interpreted as the attempt to find a physical theory that is less dualistic. However, if one rgards this dualism as resulting from the requirements of measurements, one can hope for a weakening of the dualism but not expect to remove it completely.Dedicated to Professor Hans-Jürgen Treder on his 60th birthday, September 4, 1988.     

Quantum Information in Relativity: The Challenge of QFT Measurements

Proposed quantum experiments in deep space will be able to explore quantum information issues in regimes where relativistic effects are important. In this essay, we argue that a proper extension of quantum information theory into the relativistic domain requires the expression of all informational notions in terms of quantum field theoretic (QFT) concepts. This task requires a working and practicable theory of QFT measurements. We present the foundational problems in constructing such a theory, especially in relation to longstanding causality and locality issues in the foundations of QFT. Finally, we present the ongoing Quantum Temporal Probabilities program for constructing a measurement theory that (i) works, in principle, for any QFT, (ii) allows for a first- principles investigation of all relevant issues of causality and locality, and (iii) it can be directly applied to experiments of current interest.     

Quantum Versus Stochastic Processes and the Role of Complex Numbers

We argue that the complex numbers are an irreducible object of quantum probability: this can be seen in the measurements of geometric phases that have no classical probabilistic analogue. Having the complex phases as primitive ingredient implies that we need to accept nonadditive probabilities. This has the desirable consequence of removing constraints of standard theorems about the possibility of describing quantum theory with commutative variables. Motivated by the formalism of consistent histories and keeping an analogy with the theory of stochastic processes, we develop a (statistical) theory of quantum processes: they are characterized by the introduction of a density matrix on phase space paths (it thus includes phase information) and fully reproduces quantum mechanical predictions. We can write quantum differential equations (in analogy to Langevin equation) that could be interpreted as referring to individual quantum systems. We describe the reconstruction theorem by which a quantum process can yield the standard Hilbert space structure if the Markov property is imposed. We discuss the relevance of our results for the interpretation of quantum theory (a sample space is possible if probabilities are nonadditive) and quantum gravity (the Hilbert space arises here after the consideration of a background causal structure).     

Quantum Inequality Bounds for Free Rarita-Schwinger Field in Flat Spacetime

Although quantum field theory allows the local energy density negative, it also places severe restrictions on the negative energy. One of the restrictions is the quantum energy inequality (QEI), in which the energy density is averaged over time, or space, or over space and time. By now temporal QEIs have been established for various quantum fields, but less work has been done for the spacetime quantum energy inequality. In this paper we deal with the free Rarita-Schwinger field and present a quantum inequality bound on the energy density averaged over space and time.Comparison with the QEI for the Rarita-Schwinger field shows that the lower bound is the same with the QEI. At the same time, we find the quantum inequality for the Rarita-Schwinger field is weaker than those for the scalar and Dirac fields. This fact gives further support to the conjecture that the more freedom the field has, the more easily the field displays negative energy density and the weaker the quantum inequality becomes.     

Dynamical Measurement's Uncertainty in the Curved Space‐Time

The dynamical characteristics of measurement's uncertainty are investigated under two modes of Dirac field in the Garfinkle–Horowitz–Strominger dilation space‐time. It shows that the Hawking effect induced by the thermal field would result in an expansion of the entropic uncertainty with increasing dilation‐parameter value, as the systemic quantum coherence reduces, reflecting that the Hawking effect could undermine the systemic coherence. Meanwhile, the intrinsic relationship between the uncertainty and quantum coherence is obtained, and it is revealed that the uncertainty's bound is anti‐correlated with the system's quantum coherence. Furthermore, it is illustrated that the systemic mixedness is correlated with the uncertainty to a large extent. Via the information flow theory, various correlations including quantum and classical aspects, which can be used to form a physical explanation on the relationship between the uncertainty and quantum coherence, are also analyzed. Additionally, this investigation is extended to the case of multi‐component measurement, and the applications of the entropic uncertainty relation are illustrated on entanglement criterion and quantum channel capacity. Lastly, it is declared that the measurement uncertainty can be quantitatively suppressed through optimal quantum weak measurement. These investigations might pave an avenue to understand the measurement's uncertainty in the curved space‐time.     

Physical continuum and the problem of a finitistic quantum field theory

To formulate a finitistic quantum field theory, the hypothesis is made that the continuum of space and time is countable possessing the cardinal number ₀. With the integers having the same cardinal number, it is therefore assumed that distances in space and time can be expressed only in integer multiples of a fundamental length and time. To preserve the condition of causality, a quantized field theory derived under this assumption must be expressed in absolute space and time, with the field equation invariant under Galilei transformations. It is shown that such a theory not only can be formulated in full agreement with all the postulates of quantum mechanics, but that it leads to Lorentz invariance as a dynamic symmetry in the limit of low energies. If the smallest length and time are chosen to be equal to the Planck length and time, respectively, observable departures from the predictions of special relativity would become effective only in approaching the Planck energy of 10¹⁹ GeV.