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Norbert Dragon

Publications and source records attributed to Norbert Dragon.

14 recordsLinked to original sources
arXiv2023

Relativistic Covariance of Scattering

We analyze relativistic quantum scattering in the Schr\"odinger picture. The suggestive requirement of translational invariance and conservation of the four-momentum, that the interacting Hamiltonian commute with the four-momentum $P$ of free particles, is shown to imply the absence of interactions. The relaxed requirement, that the interacting Hamiltonian $H'$ commute with the four-velocity $U= P/M$, $M=\sqrt{P^2}$, allows Poincar\'e covariant interactions just as in the nonrelativistic case. If the $S$-matrix is Lorentz invariant, it still commutes with the four-momentum $P$ though $H'$ does not. Shifted observers, whose translations are generated by the four-velocity $U$, just see a shifted superposition of near-mass-degenerate states with unchanged relative phases, while the four-momentum generates oscillated superpositions with changed relative phases.

quant-ph↗
arXiv2022

The Rough with the Smooth of the Light Cone String

The polynomials in the generators of a unitary representation of the Poincar\'e group constitute an algebra which maps the dense subspace S of smooth, rapidly decreasing wavefunctions to itself. This mathematical result is highly welcome to physicists, who previously just assumed their algebraic treatment of unbounded operators be justified. The smoothness, however, has the side effect that a rough operator R, which does not map a dense subspace of S to itself, has to be shown to allow for some other dense domain which is mapped to itself both by R and all generators. Otherwise their algebraic product, their concatenation, is not defined. Canonical quantization of the light cone string postulates operators $-i X^1$ and $P^- = (P^0 - P^z)/2$ and as their commutator the multiplicative operator R = P^1/(P^0 + P^z). This is not smooth but rough on the negative z-axis of massless momentum. Using only the commutation relations of P^m with the generators $-i M_iz$ of rotations in the $P^i$-$P^z$-plane we show that on massless states the operator $R$ is inconsistent with a unitary representation of SO(D-1). This makes the algebraic determination of the critical dimension, $D=26$, of the bosonic string meaningless: if the massless states of the light cone string admit R then they do not admit a unitary representation of the subgroup SO(D-1) of the Poincar\'e group. With analogous arguments we show: Massless multiplets are inconsistent with a translation group of the spatial momentum which is generated by a self-adjoint spatial position operator $X$.

hep-th↗
arXiv2022

Heisenberg versus the Covariant String

A Poincar\'e multiplet of mass eigenstates $\bigl(P^2 - m^2\bigr)\Psi = 0$ cannot be a subspace of a space with a $D$-vector position operator $X=(X_0,\dots X_{D-1})$: the Heisenberg algebra $[P^m, X_n] = i \delta^m{}_n$ implies by a simple argument that each Poincar\'e multiplet of definite mass vanishes. The same conclusion follows from the Stone-von Neumann theorem. In a quantum theory the constraint of an absolutely continuous spectrum to a lower dimensional submanifold yields zero even if Dirac's treatment of the corresponding classical constraint defines a symplectic submanifold with a consistent corresponding quantum model. Its Hilbert space is not a subspace of the unconstrained theory. Hence the operator relations of the unconstrained model need not carry over to the constrained model. Our argument excludes quantized worldline models of relativistic particles and the physical states of the covariant quantum string. We correct misconceptions about the generators of Lorentz transformations acting on particles.

hep-th↗
arXiv2022

Bundle Structure of Massless Unitary Representations of the Poincar\'e Group

Reviewing the construction of induced representations of the Poincar\'e group of four-dimensional spacetime we find all massive representations, including the ones acting on interacting many-particle states. Massless momentum wavefunctions of non-vanishing helicity turn out to be more precisely sections of a U(1)-bundle over the massless shell, a property which to date was overlooked in bracket notation. Our traditional notation enables questions about square integrability and smoothness. Their answers complete the picture of relativistic quantum physics. Frobenius' reciprocity theorem prohibits massless one-particle states with total angular momentum less than the modulus of the helicity. There is no two-photon state with J=1, explaining the longevity of orthopositronium. Partial derivatives of the momentum wave functions are no operators which can be applied to massless states Psi with nonvanishing helicity. They allow only for covariant, noncommuting derivatives. The massless shell has a noncommutative geometry with helicity being its topological charge. A spatial position operator for Psi which constitutes Heisenberg pairs with the spatial momentum, is excluded by the smoothness requirement of the domain of the Lorentz generators.

hep-th↗
arXiv2022

The cosmological constant as a boundary term

We compare the path integral for transition functions in unimodular gravity and in general relativity. In unimodular gravity the cosmological constant is a property of states that are specified at the boundaries whereas in general relativity the cosmological constant is a parameter of the action. Unimodular gravity with a nondynamical background spacetime volume element has a time variable that is canonically conjugate to the cosmological constant. Wave functions depend on time and satisfy a Schr\"odinger equation. On the contrary, in the covariant version of unimodular gravity with a 3-form gauge field, proposed by Henneaux and Teitelboim, wave functions are time independent and satisfy a Wheeler-DeWitt equation, as in general relativity. The 3-form gauge field integrated over spacelike hypersurfaces becomes a "cosmic time" only in the semiclassical approximation. In unimodular gravity the smallness of the observed cosmological constant has to be explained as a property of the initial state.

hep-th↗
arXiv2022

Heisenberg Algebra and String Theory

If the algebra of the Poincar\'e generators is enlarged by the spacetime position operator $X=(X_0,\dots, X_{D-1})$ then the spectra of the momentum $P$ and the mass $P^2$ are unbounded and continuous. In particular, the constraint $(P^2 - m^2)\Psi_{\text{phys}}=0$ of the covariant string has no solution in the space which admits $X$: All physical states vanish, $\Psi_{\text{phys}}=0$. Vice versa, a space spanned by mass eigenstates does not admit the position operator $X$ in $D$ dimensions. A massless particle does not allow a spatial position operator $\vec X$. The domain of Heisenberg pairs $X^i$ and $P^j$, $i,j\in \{1,\dots D-2\}$, $D > 2$, which commute with $P^+=(P^0 + P_z)/\sqrt{2}$, $[P^+,X^i] = 0$, does not allow for a space with massless or tachyonic states, which is mapped to itself by rotations, leave alone Lorentz transformations. This is true in all dimensions and makes the algebraic calculation of the critical dimension, $D=26$, of the bosonic string meaningless: the light cone string is not Lorentz invariant.

hep-th↗
arXiv2016

Currents for Arbitrary Helicity

Using Mackey's classification of unitary representations of the Poincar\'e group on massles states of arbitrary helicity we disprove the claim that states with helicity |h|>=1 cannot couple to a conserved current by constructing such a current.

hep-th↗
arXiv2012

BRST Symmetry and Cohomology

We present the mathematical considerations which determine all gauge invariant actions and anomaly candidates in gauge theories of standard type such as ordinary or gravitational Yang Mills theories. Starting from elementary concepts of field theory the discussion tries to be explicit and complete, only the cohomology of simple Lie algebras it quoted from the literature.

hep-th↗
arXiv1997

Polynomial Form of the Stueckelberg Model

The Stueckelberg model for massive vector fields is cast into a BRS invariant, polynomial form. Its symmetry algebra simplifies to an abelian gauge symmetry which is sufficient to decouple the negative norm states. The propagators fall off like $1/k^2$ and the Lagrangean is polynomial but it is not powercounting renormalizable due to derivative couplings.

hep-th↗
arXiv1996

BRS Symmetry and Cohomology

The BRS symmetry determines physical states, Lagrange densities and candidate anomalies. It renders gauge fixing unobservable in physical states and is required if negative norm states are to decouple also in interacting models. The relevant mathematical structures and the elementary cohomological investigations are presented.

hep-th↗