arXiv subjects

K. Scharnhorst

Publications and source records attributed to K. Scharnhorst.

16 recordsLinked to original sources

arXiv2023

Photon-photon scattering and related phenomena. Experimental and theoretical approaches: The early period

We review the literature on possible violations of the superposition principle for electromagnetic fields in vacuum from the earliest studies until the emergence of renormalized QED at the end of the 1940's. The exposition covers experimental work on photon-photon scattering and the propagation of light in external electromagnetic fields and relevant theoretical work on nonlinear electrodynamic theories (Born-Infeld theory and QED) until the year 1949. To enrich the picture, pieces of reminiscences from a number of (theoretical) physicists on their work in this field are collected and included or appended.

physics.hist-ph↗

arXiv2023

On self-consistency in quantum field theory

A bootstrap approach to the effective action in quantum field theory is discussed which entails the invariance under quantum fluctuations of the effective action describing physical reality (via the S-matrix).

hep-th↗

arXiv2013

Entanglement capabilities of the spin representation of (3+1)D-conformal transformations

Relying on a mathematical analogy of the pure states of the two-qubit system of quantum information theory with four-component spinors we introduce the concept of the intrinsic entanglement of spinors. To explore its physical sense we study the entanglement capabilities of the spin representation of (pseudo-) conformal transformations in (3+1)-dimensional Minkowski space-time. We find that only those tensor product structures can sensibly be introduced in spinor space for which a given spinor is not entangled.

quant-ph↗

arXiv2011

Nonlinear Bogolyubov-Valatin transformations: 2 modes

Extending our earlier study of nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for one fermionic mode, in the present paper we perform a thorough study of general (nonlinear) canonical transformations for two fermionic modes. We find that the Bogolyubov-Valatin group for n=2 fermionic modes which can be implemented by means of unitary SU(2^n = 4) transformations is isomorphic to SO(6;R)/Z_2. The investigation touches on a number of subjects. As a novelty from a mathematical point of view, we study the structure of nonlinear basis transformations in a Clifford algebra [specifically, in the Clifford algebra C(0,4)] entailing (supersymmetric) transformations among multivectors of different grades. A prominent algebraic role in this context is being played by biparavectors (linear combinations of products of Dirac matrices, quadriquaternions, sedenions) and spin bivectors (antisymmetric complex matrices). The studied biparavectors are equivalent to Eddington's E-numbers and can be understood in terms of the tensor product of two commuting copies of the division algebra of quaternions H. From a physical point of view, we present a method to diagonalize any arbitrary two-fermion Hamiltonian. Relying on Jordan-Wigner transformations for two-spin-1/2 and single-spin-3/2 systems, we also study nonlinear spin transformations and the related problem of diagonalizing arbitrary two-spin-1/2 and single-spin-3/2 Hamiltonians. Finally, from a calculational point of view, we pay due attention to explicit parametrizations of SU(4) and SO(6;R) matrices (of respective sizes 4x4 and 6x6) and their mutual relation.

math-ph↗

arXiv2005

Nonlinear Bogolyubov-Valatin transformations and quaternions

In introducing second quantization for fermions, Jordan and Wigner (1927/1928) observed that the algebra of a single pair of fermion creation and annihilation operators in quantum mechanics is closely related to the algebra of quaternions H. For the first time, here we exploit this fact to study nonlinear Bogolyubov-Valatin transformations (canonical transformations for fermions) for a single fermionic mode. By means of these transformations, a class of fermionic Hamiltonians in an external field is related to the standard Fermi oscillator.

quant-ph↗

arXiv2003

A Grassmann integral equation

The present study introduces and investigates a new type of equation which is called Grassmann integral equation in analogy to integral equations studied in real analysis. A Grassmann integral equation is an equation which involves Grassmann integrations and which is to be obeyed by an unknown function over a (finite-dimensional) Grassmann algebra G_m. A particular type of Grassmann integral equations is explicitly studied for certain low-dimensional Grassmann algebras. The choice of the equation under investigation is motivated by the effective action formalism of (lattice) quantum field theory. In a very general setting, for the Grassmann algebras G_2n, n = 2,3,4, the finite-dimensional analogues of the generating functionals of the Green functions are worked out explicitly by solving a coupled system of nonlinear matrix equations. Finally, by imposing the condition G[{\barΨ},Ψ] = G_0[{λ\barΨ}, {λΨ}] + const., 0<λ\in R (\barΨ_k, Ψ_k, k=1,...,n, are the generators of the Grassmann algebra G_2n), between the finite-dimensional analogues G_0 and G of the (``classical'') action and effective action functionals, respectively, a special Grassmann integral equation is being established and solved which also is equivalent to a coupled system of nonlinear matrix equations. If λ\not= 1, solutions to this Grassmann integral equation exist for n=2 (and consequently, also for any even value of n, specifically, for n=4) but not for n=3. If λ=1, the considered Grassmann integral equation has always a solution which corresponds to a Gaussian integral, but remarkably in the case n=4 a further solution is found which corresponds to a non-Gaussian integral. The investigation sheds light on the structures to be met for Grassmann algebras G_2n with arbitrarily chosen n.

math-ph↗

arXiv2000

Isocliny in spinor space and Wilson fermions

We show that Clifford algebras are closely related to the study of isoclinic subspaces of spinor spaces and, consequently, to the Hurwitz-Radon matrix problem. Isocliny angles are introduced to parametrize gamma matrices, i.e., matrix representations of the generators of finite-dimensional Clifford algebras C(m,n). Restricting the consideration to the Clifford algebra C(4,0), this parametrization is then applied to the study of Dirac traces occurring in Euclidean lattice quantum field theory within the hopping parameter expansion for Wilson fermions.

hep-lat↗

arXiv1999

Angles in Complex Vector Spaces

The article reviews some of the (fairly scattered) information available in the mathematical literature on the subject of angles in complex vector spaces. The following angles and their relations are considered: Euclidean, complex, and Hermitian angles, (Kasner's) pseudo-angle, the Kaehler angle (synonyms for the latter used in the literature are: angle of inclination, characteristic deviation, holomorphic deviation, holomorphy angle, Wirtinger angle, slant angle).

math.HO↗

arXiv1998

A special irreducible matrix representation of the real Clifford algebra C(3,1)

4x4 Dirac (gamma) matrices (irreducible matrix representations of the Clifford algebras C(3,1), C(1,3), C(4,0)) are an essential part of many calculations in quantum physics. Although the final physical results do not depend on the applied representation of the Dirac matrices (e.g. due to the invariance of traces of products of Dirac matrices), the appropriate choice of the representation used may facilitate the analysis. The present paper introduces a particularly symmetric real representation of 4x4 Dirac matrices (Majorana representation) which may prove useful in the future. As a byproduct, a compact formula for (transformed) Pauli matrices is found. The consideration is based on the role played by isoclinic 2-planes in the geometry of the real Clifford algebra C(3,0) which provide an invariant geometric frame for it. It can be generalized to larger Clifford algebras.

hep-th↗

arXiv1998

The velocities of light in modified QED vacua

QED vacua under the influence of external conditions (background fields, finite temperature, boundary conditions) can be considered as dispersive media whose complex behaviour can no longer be described in terms of a single universal vacuum velocity of light c. Beginning in the early 1950's (J.S. Toll), quantum field theoretic investigations have led to considerable insight into the relation between the vacuum structure and the propagation of light. Recent years have witnessed a significant growth of activity in this field of research. After a short overview, two characteristic situations are discussed: the propagation of light in a constant homogeneous magnetic field and in a Casimir vacuum. The latter appears to be particularly interesting because the Casimir vacuum has been found to exhibit modes of the propagation of light with phase and group velocities larger than c in the low frequency domain omega<<m where m is the electron mass. The impact of this result on the front velocity of light in a Casimir vacuum is discussed by means of the Kramers-Kronig relation.

hep-th↗

arXiv1998

$O(α)$ Radiative Correction to the Casimir Energy for Penetrable Mirrors

The leading radiative correction to the Casimir energy for two parallel penetrable mirrors is calculated within QED perturbation theory. It is found to be of the order $α$ like the known radiative correction for ideally reflecting mirrors from which it differs only by a monotonic, powerlike function of the frequency at which the mirrors become transparent. This shows that the $O(α^2)$ radiative correction calculated recently by Kong and Ravndal for ideally reflecting mirrors on the basis of effective field theory methods remains subleading even for the physical case of penetrable mirrors.

hep-th↗

arXiv1996

The exact equivalence of the one-flavour lattice Thirring model with Wilson fermions to a two-colour loop model

Within Euclidean lattice field theory an exact equivalence between the one-flavour 2D Thirring model with Wilson fermions and Wilson parameter $r = 1$ to a two-colour loop model on the square lattice is established. For non-interacting fermions this model reduces to an exactly solved loop model which is known to be a free fermion model. The two-colour loop model equivalent to the Thirring model can also be understood as a 4-state 49-vertex model.

hep-lat↗

arXiv1996

The exact equivalence of the two-flavour strong coupling lattice Schwinger model with Wilson fermions to a vertex model

In this paper a method previously employed by Salmhofer to establish an exact equivalence of the one-flavour strong coupling lattice Schwinger model with Wilson fermions to some 8-vertex model is applied to the case with two flavours. As this method is fairly general and can be applied to strong coupling QED and purely fermionic models with any (sufficiently small) number of Wilson fermions in any dimension the purpose of the present study is mainly a methodical one in order to gain some further experience with it. In the paper the vertex model equivalent to the two-flavour strong coupling lattice Schwinger model with Wilson fermions is found. It turns out to be some modified 3-state 20-vertex model on the square lattice, which can also be understood as a regular 6-state vertex model. In analogy with the one- flavour case, this model can be viewed as some loop model.

hep-lat↗

arXiv1995

Some approximate analytical methods in the study of the self-avoiding loop model with variable bending rigidity and the critical behaviour of the strong coupling lattice Schwinger model with Wilson fermions

Some time ago Salmhofer demonstrated the equivalence of the strong coupling lattice Schwinger model with Wilson fermions to a certain 8-vertex model which can be understood as a self-avoiding loop model on the square lattice with bending rigidity $η= 1/2$ and monomer weight $z = (2κ)^{-2}$. The present paper applies two approximate analytical methods to the investigation of critical properties of the self-avoiding loop model with variable bending rigidity, discusses their validity and makes comparison with known MC results. One method is based on the independent loop approximation used in the literature for studying phase transitions in polymers, liquid helium and cosmic strings. The second method relies on the known exact solution of the self-avoiding loop model with bending rigidity $η= 1/\sqrt{2}$. The present investigation confirms recent findings that the strong coupling lattice Schwinger model becomes critical for $κ_{cr} \simeq 0.38-0.39$. The phase transition is of second order and lies in the Ising model universality class. Finally, the central charge of the strong coupling Schwinger model at criticality is discussed and predicted to be $c = 1/2$.

hep-lat↗

arXiv1993

A Functional Integral Equation for the Complete Effective Action in Quantum Field Theory

Based on a methodological analysis of the effective action approach certain conceptual foundations of quantum field theory are reconsidered to establish a quest for an equation for the effective action. Relying on the functional integral formulation of Lagrangian quantum field theory a functional integral equation for the complete effective action is proposed which can be understood as a certain fixed point condition. This is motivated by a critical attitude towards the distinction artificial from an experimental point of view between classical and effective action. While for free field theories nothing new is accomplished, for interacting theories the concept differs from the established paradigm. The analysis of this new concept is concentrated on gauge field theories treating QED as the prototype model. An approximative approach to the functional integral equation for the complete effective action of QED is exploited to obtain certain nonperturbative information about the quadratic kernels of the action. As particular application the approximative calculation of the QED coupling constant $α$ is explicitly studied. It is understood as one of the characteristics of a fixed point given as a solution of the functional integral equation proposed. Finally, within the present approach the vacuum energy problem is considered and possible implications on the induced gravity concept are contemplated.

hep-th↗

arXiv1993

Nonlocal QED admits a finitely induced gauge field action

The Letter reconsiders a result obtained by Chrétien and Peierls in 1954 within nonlocal QED in 4D [Proc. Roy. Soc. London A 223, 468]. Starting from secondly quantized fermions subject to a nonlocal action with the kernel $[ i\not\partial_x a(x) - m b(x)]$ and gauge covariantly coupled to an external U(1) gauge field they found that for $a = b$ the induced gauge field action cannot be made finite irrespectively of the choice of the nonlocality $a$ $(= b)$. But, the general case $a \neq b$ naturally to be studied admits a finitely induced gauge field action, as the present Letter demonstrates.

hep-th↗