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Maciej Trzetrzelewski

Publications and source records attributed to Maciej Trzetrzelewski.

At least 19 recordsLinked to original sources
arXiv2018

On the Equivalence Principle and Electrodynamics of Moving Bodies

We consider a certain extension of the Einstein's elevator thought experiment by assuming that the elevator is charged and falls into an electromagnetic field. We argue, on grounds of the Equivalence Principle, that an observer-dependent metric should exist, for which the geodesics coincide with the trajectories of the charged body in the electromagnetic field. We give a solution to this problem by finding such a metric and hence we are able to recover the Lorentz-force law. Recovery of the leading terms of the radiation reaction force is also possible with this approach.

physics.gen-ph↗
arXiv2018

Dirac equation for strings

Starting with a Nambu-Goto action, a Dirac-like equation can be constructed by taking the square-root of the momentum constraint. The eigenvalues of the resulting Hamiltonian are real and correspond to masses of the excited string. In particular there are no tachyons. A special case of radial oscillations of a closed string in Minkowski space-time admits exact solutions in terms of wave functions of the harmonic oscillator.

hep-th↗
arXiv2018

Relativistic Black-Scholes model

Black-Scholes equation, after a certain coordinate transformation, is equivalent to the heat equation. On the other hand the relativistic extension of the latter, the telegraphers equation, can be derived from the Euclidean version of the Dirac equation. Therefore the relativistic extension of the Black-Scholes model follows from relativistic quantum mechanics quite naturally. We investigate this particular model for the case of European vanilla options. Due to the notion of locality incorporated in this way one finds that the volatility frown-like effect appears when comparing to the original Black-Scholes model.

q-fin.PR↗
arXiv2015

On the mass term of the Dirac equation

We consider the generalization of the Dirac equation where the mass term is an arbitrary matrix M. A general form of M, consistent with the mass shell constraint, is derived and proven to be equivalent to the original Dirac equation.

hep-th↗
arXiv2013

Dirac equation for embedded 4-geometries

We apply Dirac's square root idea to constraints for embedded 4-geometries swept by a 3-dimensional membrane. The resulting Dirac-like equation is then analyzed for general coordinates as well as for the case of a Friedmann-Robertson-Walker metric for spatially closed geometries. The problem of the singularity formation at quantum level is addressed.

hep-th↗
arXiv2012

Towards the Ground State of the Supermembrane

The explicit, near the origin, form of the ground state of the SU(2) supermembrane matrix model is studied. We evaluate the 2nd order terms of the Taylor expansion of the wave-function, which together with the 0th and the 1st order terms can be used to determine other terms by recurrence equations coming from the Schroedinger equation.

hep-th↗
arXiv2012

A note on the principle of least action and Dirac matrices

Many Lagrangians of physical theories can be expressed as eigenvalues of certain, relatively simple, matrices involving Dirac gamma matrices. We give concrete examples for Lagrangian corresponding to a point particle coupled to electromagnetic field, electrodynamics, nonabelian gauge theories, extended objects and gravity. We also discuss (in case of a point particle) what are the implications of the least action principle applied to matrix Lagrangians.

math-ph↗
arXiv2012

Matrix regularization of embedded 4-manifolds

We consider products of two 2-manifolds such as S^2 x S^2, embedded in Euclidean space and show that the corresponding 4-volume preserving diffeomorphism algebra can be approximated by a tensor product SU(N)xSU(N) i.e. functions on a manifold are approximated by the Kronecker product of two SU(N) matrices. A regularization of the 4-sphere is also performed by constructing N^2 x N^2 matrix representations of the 4-algebra (and as a byproduct of the 3-algebra which makes the regularization of S^3 also possible).

math-ph↗
arXiv2011

Dirac equation for membranes

Dirac's idea of taking the square root of constraints is applied to the case of extended objects concentrating on membranes in D=4 space-time dimensions. The resulting equation is Lorentz invariant and predicts an infinite hierarchy of positive and negative masses (tension). There are no tachyonic solutions.

hep-th↗
arXiv2011

Dirac membrane and hadrons

In c=1 units the product (mass x radius) for the neutron and the proton is about 4.7\hbar assuming their radii equal to 1fm. We show that the corresponding products for the Dirac neutral and charged membrane coincide and are equal 1.6\hbar.

hep-th↗
arXiv2011

Lorentz-invariant membranes and finite matrix approximations

The question of Lorentz invariance for finite N approximations of relativistic membranes is addressed. We find that one of the classical manifestations of Lorentz-invariance is not possible for NxN matrices (at least when N=2 or 3). How the symmetry is restored in the large N limit is studied numerically.

hep-th↗
arXiv2009

Spiky membranes

We study spiky configurations of membranes in the SO(d)xSU(N) invariant matrix models. A class of exact solutions (analogous to plane-waves) of the corresponding Schroedinger equation for an arbitrary N is discussed. If the large N limit is performed so that the energy scales like N^2, the N=infinity wavefunctions reduce to the ground state of the d-dimensional harmonic oscillator.

hep-th↗
arXiv2008

Construction of the Zero-Energy State of SU(2)-Matrix Theory: Near the Origin

We explicitly construct a (unique) Spin(9)xSU(2) singlet state, phi, involving only the fermionic degrees of freedom of the supersymmetric matrix-model corresponding to reduced 10-dimensional super Yang-Mills theory, resp. supermembranes in 11-dimensional Minkowski space. Any non-singular wavefunction annihilated by the 16 supercharges of SU(2) matrix theory must, at the origin (where it is assumed to be non-vanishing) reduce to phi.

hep-th↗
arXiv2008

Spin(9) Average of SU(N) Matrix Models I. Hamiltonian

We apply a method of group averaging to states and operators appearing in (truncations of) the Spin(9) x SU(N) invariant matrix models. We find that there is an exact correspondence between the standard supersymmetric Hamiltonian and the Spin(9) average of a relatively simple lower-dimensional model.

hep-th↗
arXiv2008

Octonionic twists for supermembrane matrix models

A certain G_2 times U(1) invariant Hamiltonian arising from the standard membrane matrix model via conjugating any of the supercharges by a cubic, octonionic, exponential is proven to have a spectrum covering the whole positive half-axis. The model could be useful in determining a normalizable zero-energy state in the original SO(9) invariant SU(N) matrix model.

hep-th↗
arXiv2008

Supergravitons from one loop perturbative N=4 SYM

We determine the partition function of 1/16 BPS operators in N=4 SYM at weak coupling at the one-loop level in the planar limit. This partition function is significantly different from the one computed at zero coupling. We find that it coincides precisely with the partition function of a gas of 1/16 BPS `supergravitons' in AdS_5xS^5.

hep-th↗
arXiv2007

The number of gauge singlets in supersymmetric Yang-Mills quantum mechanics

We compute generating functions for number of U(N) (SU(N)) singlets in Fock space in several space dimensions. The motivation to find the explicit form of the functions is from the numerical approach to supersymmetric Yang-Mills quantum mechanics, based on Fock space. Incidently the functions give many important insights into the quantum mechanical models based U(N) (SU(N)) gauge group.

hep-th↗