arXiv subjects
Publications and source records attributed to Jaroslaw Wawrzycki.
We present a rigorous construction of the Feynman integral on the compactified Einstein Universe (EU) using white noise calculus. Presented construction of the functional averaging may also be thought of as a solution of the problem posed by Bogoliubov and Shirkov in Chap.VIII.43 of their book Introduction to the Theory of Quantized Fields'', Wiley 1980.
This work is devoted to the causal perturbative Quantum Field Theory (QFT) due to Bogoliubov, including QED and other realistic QFT. It is given the white noise formulation of this theory. The white noise analysis and the Hida operators as the creation and annihilation operators for free fields are used. The whole Bogoliubov method is unchanged. Causal axioms of such QFT make sense on any globally causal space-times. Perturbative QFT with Hida operators is analysed on the flat Minkowski space-time and on the static Einstein Universe (EU). On the flat Minkowski space-time this allowed us to go a step further in the analysis of the adiabatic limit problem, the existence of which has a much wider scope than in the theory based on Wightman's operator valued distributions. The natural condition of existence of the adiabatic limit (as generalized integral kernel operators of white noise calculus) for higher-order contributions to the interacting fields, together with the remaining assumptions of causal QFT (including the natural invariance conditions, e.g. gauge invariance), allowed us to reduce the freedom in choice of renormalization and imposed nontrivial conditions on the masses of elementary particles. Obtained results on the flat space-time are confirmed by the analogous results obtained for QFT on UE, where, moreover, the operators of the interacting fields are much more regular, so, e.g. a consequent treatment of the bound state problem becomes possible on EU. Our approach also enables the analysis of the infrared asymptotics of QED on flat Minkowski spacetime, which we also present.
We present the Bogoliubov's causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, \emph{i.e.} for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared -- and ultraviolet -- divergences in the transition to the adiabatic limit for interacting fields. Here we present the inductive step. We are especially concentrated with the analysis of the second order contribution to the scattering operator, and give explicit computations using the calculus of integral kernel operators, \emph{i.e} contractions of their vector-valued kernels, and splittings of the contractions into advanced and retarded parts. Explicit example of QED is provided, together with the detailed analysis of the key point in avoiding UV divergence.
We present the Bogoliubov's causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, \emph{i.e.} for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared -- and ultraviolet -- divergences in the transition to the adiabatic limit for interacting fields and elimination of the free parameters of the theory, associated with the choice of normalization in computation of the retarded and advanced parts of causal distributions (corresponding to the freedom in the choice of renormalization scheme). This enhances the predictive power of the theory, and in particular allows to derive non-trivial mass relations. The approach is general and can be applied to investigate any perturbative QFT.
We present the Bogoliubov's causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, \emph{i.e.} for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared -- and ultraviolet -- divergences in the transition to the adiabatic limit for interacting fields. We present here the analysis of the causal axioms for the scattering operator with the Hida operators as the creation-annihilation operators.
We present the Bogoliubov's causal perturbative QFT, which includes only one refinement: the creation-annihilation operators at a point, i.e. for a specific momentum, are mathematically interpreted as the Hida operators from the white noise analysis. We leave the rest of the theory completely unchanged. This allows avoiding infrared -- and ultraviolet -- divergences in the transition to the adiabatic limit for interacting fields. We present here existence proof of the adiabatic limit for interacting fields in causal QED with Hida operators. This limit exists if and only if the normalization in the Epstein-Glaser splitting of the causal distributions, in the construction of the scattering operator, is "natural", and thus eliminates arbitrariness in the choice of the splitting making the theory definite, with its predictive power considerably strengthened. For example, we present a charge-mass relation which can be proved within this theory, and which is confirmed experimentally.
We present a proof of positivity of an invariant kernel, which is of basic importance for the Staruszkiewicz theory of the quantum Coulomb field. Presented proof of positivity is independent of the Staruszkiewicz theory and is based on the classical Schoenberg's theorem for conditionally negative definite functions, as well as on the generalized Bochner's theorem.
We will present the axioms of Bogoliubov's causal perturbative QFT in which the creation-annihilation operators are interpreted as Hida operators. We will shortly present the results that can be achieved in this theory: 1. Removal of UV and IR infinity in the scattering operator, 2. Existence of the adiabatic limit for interacting fields in QED, 3. Proof that charged particles have non-zero mass, 4. Existence of infrared and ultraviolet asymptotics for QED.
In this work we extend the Mackey's theory of induced unitary representations on a wide class of Krein-isometric induced representations in Krein spaces. The subgroup theorem and the Kronecker product theorem are shown to be valid for the induced representations of this class. Among the class of representations which are subsumed by this extension there are all the representations acting in the single particle Krein-Hilbert space and in Fock-Krein spaces of the mass less gauge free fields underling the Standard Model in the local gauge BRST formalutaion, e. g. of the electromagnetic potential field.
At the classical level the electromagnetic field can be well identified at the spatial infinity. Staruszkiewicz pointed out that the quantization of the electromagnetic field at spatial infinity is essentially unique and follows from the two fundamental principles: 1) gauge invariance and 2) canonical commutation relations for canonically conjugated generalized coordinates, and constructed a simple and mathematically transparent quantum theory of the Coulomb field, predicting (among other things) a relation between the theory of unitary representations of the $SL(2, \mathbb{C})$ group and the fine structure constant. Until now this theory has stayed outside the main stream of the perturbative development in QED, mainly due to the unsolved infra-red-type (IR) problems in the perturbative approach. Recently however there has been performed a more careful analysis of free fields including the mass less free gauge fields, such as the electromagnetic potential field, their Wick and chronological products, which revealed the need for a more careful and white noise construction of these fields, and which opened a way to resolve IR problems (at least those which shows up at each order separately). Comparison of the perturbatively constructed field at spatial infinity with the quantum phase field of the Staruszkiewicz theory leads to the proof of universality of the unit of charge.
We present in this paper how the single-photon wave function for transversal photons (with the direct sum of ordinary unitary representations of helicity 1 and -1 acting on it) is subsumed within the formalism of Gupta-Bleuler for the quantized free electromagnetic field in the Krein space (i.e. in the ordinary Hilbert space endowed with the Gupta-Bleuler operator $η$). Rigorous Gupta-Bleuler quantization of the free electromagnetic field is based on a generalization of ours (published formerly) of the Mackey theory of induced representations which includes representations preserving the indefinite Krein inner-product given by the Gupta-Bleuler operator and acting in the Krein space. The free electromagnetic field is constructed by application of the Segal's second quantization functor to the specific Krein-isometric representation. A closed subspace $\mathcal{H}_{\textrm{tr}}$ of the single-photon Krein space on which the indefinite Krein-inner-product is strictly positive is constructed such that the Krein-isometric single-photon representation generates modulo unphysical states precisely the action of a representation which preserves the positive inner product on $\mathcal{H}_{\textrm{tr}}$ induced by the Krein inner product, and is equal to the direct sum of ordinary unitary representations of helicity 1 and -1 respectively. Two states of single photon Krein space are physically equivalent whenever differ by a state of Krein norm zero and whose projection on $\mathcal{H}_{\textrm{tr}}$, in the sense of the Krein-inner-product, vanishes. In particulart it follows that the results of Białynicki-Birula on the single-photon wave function may be reconciled with the micro-local perturbative approach to QED initiated by Stückelberg and Bogoliubov.
Using the results obtained by Staruszkiewicz in Acta Phys. Pol. B 23, 591 (1992) and in Acta Phys. Pol. B 23, 927 (1992) we show that the representations acting in the eigenspaces of the total charge operator corresponding to the eigenvalues $n_1, n_2$ whose absolute values are less than or equal $\sqrt{π/e^2}$ are inequivalent if $|n_1| \neq |n_2|$ and contain the supplementary series component acting as a discrete component. On the other hand the representations acting in the eigenspaces corresponding to eigenvalues whose absolute values are greater than $\sqrt{π/e^2}$ are all unitarily equivalent and do not contain any supplementary series component.
The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: 1) local algebraic formulation of Haag, 2) Wightman formulation and 3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between 1) and 3) and utilize the known relationships between 1) and 2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of Dütsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index -- an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.
Newton in his Principia gives an ingenious generalization of the Hellenistic theory of ratios and inspired experimentally gives a tensor-like definition of multiplication of quantities measured with his ratios. An extraordinary feature of his definition is generality: namely his definition a priori allows non commutativity of multiplication of measured quantities, which may give a non-trivial linkage to experimental facts subject to quantum mechanics discovered some two hundred years later. Mathematical scheme he introduces with this ingenious definition is closely related to the contemporary approach in spectral geometry. His definition reveals in particular that commutativity of the multiplication of quantities with physical dimension has the status of experimental assumption and does not have to be fulfilled in reality, although neither the mathematical tools nor experimental evidence could allow Newton to carry out the case when the multiplication is noncommutative. We present a detailed analysis of his definition as well as a linkage to the theory of representations of algebras with involution (and with normal forms of von Neumann algebras and Jordan Banach algebras).
In this paper we will analyze the the status of gauge freedom in quantum mechanics (QM) and quantum field theory (QFT). Along with this analysis comparison with ordinary QFT will be given. We will show how the gauge freedom problem is connected with the spacetime coordinates status - the very point at which the difficulties of QM begin. A natural solution of the above mentioned problem will be proposed in which we give a slightly more general form of QM and QFT (in comparison to the ordinary QFT) with noncommutative structure of spacetime playing fundamental role in it. We achieve it by reinterpretation of the Bohr's complementarity principle on the one hand and by incorporation of our gauge freedom analysis on the other. We will present a generalization of the Bargmann's theory of exponents of ray representations. It will be given an example involving time dependent gauge freedom describing non-relativistic quantum particle in nonrelativistic gravitational field. In this example we infer the most general Schroedinger equation and prove equality of the (passive) inertial and the gravitational masses of quantum particle.
In this announcement we generalize the Markov-Kakutani fixed point theorem for abelian semi-groups of affine transformations extending it on some class of non-commutative semi-groups. As an interesting example we apply it obtaining a generalization of the invariant version of the Hahn-Banach theorem.
We investigate the interaction of the gravitational field with a quantum particle. First, we give the proof of the equality of the inertial and the gravitational mass for the nonrelativistic quantum particle, independently of the equivalence principle. Second, we show that the macroscopic body cannot be described by the many-particle Quantum Mechanics. As an important tool we generalize the Bargmann's theory of ray representations and explain the connection with the state vector reduction problem. The Penrose's hypothesis is discussed, i.e. the hypothesis that the gravitational field may influence the state vector reduction.
We investigate the interaction of the gravitational field with a quantum particle. We derive the wave equation in the curved galilean spacetime from the very broad Quantum mechanical assumptions and from covariance under the Milne group. The inertial and gravitational masses are equal in that equation. So, we give the proof of the equality for the non-relativistic quantum particle, without applying the equivalence principle to the Schrödinger equation and witout imposing any relation to the classical equations of motion. This result constitutes a substantial strengthening of the previous result obtained by Herdegen and the author.