Remarks on Causality in Relativistic Quantum Field Theory

It is shown that the correlations predicted by relativistic quantum field theory in locally normal states between projections in local von Neumann algebras (V ₁),(V ₂) associated with spacelike separated spacetime regions V ₁,V ₂ have a (Reichenbachian) common cause located in the union of the backward light cones of V ₁ and V ₂. Further comments on causality and independence in quantum field theory are made. Originally published in International Journal of Theoretical Physics, Vol. 44, No. 7, 2005,Due to a publishing error, authorship of the article was credited incorrectly. The corrected article is reprinted in its entirety here. The online version of the original article can be found at     

Exchangeability and Conditionally Identical Common Cause Systems

A pair (A, B) of events in a classical probability measure space (Ω, p) is called exchangeable iff p(A ) = p( B). Conditionally identical common cause system of size n for the correlation is an n-partition of Ω such that (i) any member of the partition screens the correlation off and (ii) for any member {C _i } _{i ∊ I} of the partition p(A|C _i ) = p(B|C _i ). The common cause system is called proper if p(A|C _i )≠(A|C _j ) for some i ≠ j. In the paper it is shown that exchangeable correlations be explained by proper conditionally identical common cause systems in the following sense. (i) Given a proper conditionally identical common cause system of size n for the two events A and B in Ω, then the pair (A, B) will be an exchangeable (positively) correlating pair. (ii) Given any exchangeable (positively) correlating pair of events in Ω and given any finite number n > 2, then the probability space can be embedded into a larger probability space in such a way that the larger space contains a proper conditionally identical common cause system of size n for the correlation.     

Local Primitive Causality and the Common Cause Principle in Quantum Field Theory

If (V) is a net of local von Neumann algebras satisfying standard axioms of algebraic relativistic quantum field theory and V ₁ and V ₂ are spacelike separated spacetime regions, then the system ( (V ₁ ), (V ₂ ), ) is said to satisfy the Weak Reichenbach's Common Cause Principle iff for every pair of projections A (V ₁ ), B (V ₂ ) correlated in the normal state there exists a projection C belonging to a von Neumann algebra associated with a spacetime region V contained in the union of the backward light cones of V ₁ and V ₂ and disjoint from both V ₁ and V ₂ , a projection having the properties of a Reichenbachian common cause of the correlation between A and B. It is shown that if the net has the local primitive causality property then every local system ( (V ₁ ), (V ₂ ), ) with a locally normal and locally faithful state and suitable bounded V ₁ and V ₂ satisfies the Weak Reichenbach's Common Cause Principle.     

Normal and antinormal ordering of field operators in quantum optics

The relations between the probability distributionp n _vt of the number of photonsn _vt in a volumeV at timet and the probability distributions P_N(W) and P_A(W) of the integrated intensityW related to the normal and antinormal ordering of field operators and the relations between corresponding moments are studied for a field containingM modes. As the normally ordered correlations are measured with photodetectors and the antinormally ordered correlations are measured with quantum counters the relations betweenn _vt , P_N(W) and P_A(W) permit the statistical behaviour of light to be determined from measurements with photodetectors and quantum counters. The results obtained here are used for the superposition of coherent and thermal fields. It is also shown that the antinormal correlations depend explicitly on the number of modes and that in the classical limit, when the average photon occupation number per mode becomes large, the distributionsn _vt , P_N(W) and P_A(W) become equal.     

Associativity of Quantum Multiplication

We proved the associativity of the multiplication of quantum cohomology for a monotone compact symplectic manifold V for which c ₁(A)>1 for any effective class . The same proof also works for any positive compact symplectic manifold with c ₁(A)>1. Received: 17 November 1994 / Accepted: 8 May 1997     

The Multi-form Generalised Gleason Theorem

Arising from investigations of decoherence functionals in the “histories” approach to quantum mechanics, the following result is given by the methods of [14]. Let A ₁ and A ₂ be von Neumann algebras without Type I ₂ direct summands and let P(E _j ), (j=1,2) be their lattices of projections. Let m\colon; P(A ₁)×(P(A ₂)→ℂ be a bounded quantum bi-measure. Then there is a unique bounded bilinear functional M on A ₁×A ₂ which extends m. In this note we use a different approach to establish a generalisation of this result to k-fold, vector valued, quantum multi-measures. This tool is needed for further investigations of decoherence functionals in quantum theory. Received: 10 April 1998 / Accepted: 17 April 1998     

Free products of compact quantum groups

We construct and study compact quantum groups from free products ofC ^*-algebras. In this connection, we discover two mysterious classes of natural compact quantum groups,A _u (m) andA _o (m). TheA _u (m)'s (respectivelyA _o (m)'s) are non-isomorphic to each other for differentm's, and are not obtainable by the ordinary quantization method. We also clarify some basic concepts in the theory of compact quantum groups.     

Realizations of causal manifolds by quantum fields

Quantum mechanical operators and quantum fields are interpreted as realizations of timespace manifolds. Such causal manifolds are parametrized by the classes of the positive unitary operations in all complex operations, i.e., by the homogenous spacesD(n)=GL(C _R ⁿ )/U(n) withn=1 for mechanics andn=2 for relativistic fields. The rankn gives the number of both the discrete and continuous invariants used in the harmonic analysis, i.e., two characteristic masses in the relativistic case. ‘Canonical’ field theories with the familiar divergencies are inappropriate realizations of the real 4-dimensional causal manifoldD(2). Faithful timespace realizations do not lead to divergencies. In general they are reducible, but nondecomposable—in addition to representations with eigenvectors (states, particle), they incorporate principal vectors without a particle (eigenvector) basis as exemplified by the Coulomb field. In theorthogonal andunitary groupsO(N ₊,N ₋), respectively, thepositive orthogonal and unitary ones areO(N) andU(N), respectively.     

Spectra,eigenvectors and overlap functions for representation operators ofq-deformed algebras

Operators of representations corresponding to symmetric elements of theq-deformed algebrasU _q (su_1,1),U _q (so_2,1),U _q (so_3,1),U _q (so _n ) and representable by Jacobi matrices are studied. Closures of unbounded symmetric operators of representations of the algebrasU _q (su_1,1) andU _q (so_2,1) are not selfadjoint operators. For representations of the discrete series their deficiency indices are (1,1). Bounded symmetric operators of these representations are trace class operators or have continuous simple spectra. Eigenvectors of some operators of representations are evaluated explicitly. Coefficients of transition to eigenvectors (overlap coefficients) are given in terms ofq-orthogonal polynomials. It is shown how results on eigenvectors and overlap coefficients can be used for obtaining new results in representation theory ofq-deformed algebras.     

Le Groupe Quantique Compact Libre U(n)

The free analogues of U(n) in Woronowicz' theory [Wo2] are the compact matrix quantum groups introduced by Wang and Van Daele. We classify here their irreducible representations. Their fusion rules turn to be related to the combinatorics of Voiculescu's circular variable. If we find an embedding , where A _o (F) is the deformation of SU(2) studied in [B2]. We use the representation theory and Powers' method for showing that the reduced algebras A _u (F) _red are simple, with at most one trace. Received: 1 March 1996 / Accepted: 4 April 1997     

The quantum group structure associated with non-linearly extended Virasoro algebras

Recently, an infinite family of chiral Virasoro vertex operators, whose exchange algebra is given by the universalR-matrix ofSL(2) _q , has been constructed. In the present paper, the case of non-linearly (W-) extended Virasoro symmetries, related to the algebrasA _N,N>1, is considered along the same line. Contrary to the previous case (A ₁) the standardR-matrix ofSL(N+1)q does not come out, and a different solution of Yang and Baxter's equations is derived. The associated quantum group structure is displayed.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud     

Measurement of the form-factor ratios for

The form factor ratios r_V=V(0)/A₁(0), r₂=A₂(0)/A₁(0) and r₃=A₃(0)/A₁(0) in the decay , have been measured using data from charm hadroproduction experiment E791 at Fermilab. From 3034 (595) signal (background) events in the muon channel, we obtain r_V=1.84±0.11±0.09, r₂=0.75±0.08±0.09 and, as a first measurement of r₃, we find 0.04±0.33±0.29. The values of the form factor ratios r_V and r₂ measured for the muon channel are combined with the values of r_V and r₂ that we have measured in the electron channel. The combined E791 results for the muon and electron channels are r_V=1.87±0.08±0.07 and r₂=0.73±0.06±0.08.     

The q-boson realizations of the quantum groups Uq(B2) and Uq(C2)

We give explicit realization of formulae of canonical realization for the quantum enveloping algebrasU _q (B ₂)~U _q (so(5)) andU _q (C ₂)~U _q (sp(4)). In these formulae the generators of the algebra are expressed by means of 3 canonicalq-boson pairs and one auxiliary representation ofU _q (gl(2)).     

Model-independent analysis of the neutral-current interaction in the inclusive neutrino reactions

On the basis of some general assumptions on the deep inelastic structure functions, such as scaling and chiral symmetry we determine the values of {(H _V ³)²+(H _A ³)²}+η{(H _V ⁰)²+(H _A ⁰)²} andH _V ³ H _A ³+ηH _V ⁰ H _A ⁰ whereH _V ^3,0 andH _A ^3,0 are the four coupling constants characterizing the hadronic neutral current andη is the ratio of the isoscalar to isovector structure functions. General expressions are given for the kinematical averages 〈v〉 and 〈Q ²〉 for the neutral-current reactions in terms of the coupling constants. This analysis does not depend on the validity of the quark-parton model.     

Electromagnetic form factors of nucleons in a relativistic square-root potential model of independent quarks

The nucleon electromagnetic form factorsG _E ^P (q²),G _M ^P (q²) and the axial-vector form factor G_A(q²) are studied in a relativistic model of independent quarks confined by an equally mixed scalar-vector square root potentialV _q(r)=1/2(1+γ ⁰)(ar ^1/2+ν ₀) taking into account the appropriate centre-of-mass corrections. The respective root-mean-square radii associated withG _E ^P (q²) and G _A (q²) come out as [〈r ²〉 _E ^P ]^1/2=0.86 fm and 〈r _A ² 〉^1/2=0.88 fm. Restoration of chiral symmetry in this model is discussed to derive the pion-nucleon form factorG _πNN(q²) and consequently the pion-nucleon coupling constant is obtained asg _πNN(q²)=12.81 as compared tog _πNN(q²)_exp⋍13.     

Hypervirial perturbation theory for eigenenergies for two types of atomic potentials

Eigenenergies are calculated for the potentialsV ₁(r)=−(a/r)[1+(1+br)e^−2br ] andV ₂(r)=−(v/r)[1 −λr(1−Z ⁻¹)(1+λr)⁻¹], using renormalized series technique. Accurate results produced here for various eigenstates agree with those available in the literature.     

Is a Bell-type inequality for nondicotomic observables a good test of quantum mechanics?

It is enquired whether a Bell-type inequality for nondicotomic observables established in a preceding paper can be a suitable tool for testing the prediction of quantum mechanics (Q.M.) concerning the correlations s _A ·â s _B ·b observable in a decay of an unpolarized particle of arbitrary spin into two massive particlesA andB of spinss _A,s _B, respectively. It is found that such correlations cannot violate the inequality for spins of the subsystems larger than 1/2 and suggested that, apart from the possibility of finding different sensitive observables, feasible and real tests of QM against local hidden variable theories will probably always imply photon processes, owing to the simplification introduced in the density matrix of the final state by the dicotomic nature of the photon spin variables.     

Symmetries induced by conserved vector currents in the theory of a scalar field

Let us consider a quantum theory of one scalar, real, local, Poincaré covariant fieldA(x) with the restricted spectrum condition (massive one particle states and a unique vacuum). The asymptotic fieldsA _{in out} (x) are assumed to be irreducible. Our conjecture is that under some technical assumptions the charge of every real, hermitean, locally conserved, Poincaré covariant quantum (pseudo) vector fieldj (x) relatively local toA(x), appearing in this theory-vanishes. This means that in a theory of one scalar, real field with a massive particle one can not expect to get symmetry groups induced by conserved (pseudo) vector currents, only by global, selfadjoint, Poincaré invariant generators.Our arguments can be easily extended to a theory of one complex scalar field, in this case the only symmetry transformation induced by a current can be the gauge transformation.We prove also that under very weak assumptions two fields related to each other by a unitary (or similarity) transformation are equal barring some patological cases.     

Attempt to Resolve the EPR-Bell Paradox via Reichenbach's Concept of Common Cause

Reichenbach's common cause principle claims that if there is correlation betweentwo events and none of them is directly causally influenced by the other, thenthere must exist a third event that can, as a common cause, account for thecorrelation. The EPR-Bell paradox consists in the problem that we observecorrelations between spatially separated events in the EPR experiments whichdo not admit common-cause-type explanation, and it must therefore be concludedthat, contrary to relativity theory, in the realm of quantum physics there existsaction at a distance, or at least superluminal causal propagation is possible; thatis, either relativity theory or Reichenbach's common cause principle fails.By means of closer analyses of the concept of common cause and a more precisereformulation of the EPR experimental scenario, I sharpen the conclusion wecan draw from the violation of Bell's inequalities.