436 ABSTRACTS OF PAPERS TO APPEAR IN FUTURE ISSUES nucleon parity-violating couplings, may be determinable from such experiments. Further, it is possible to check the experiment of Lobashov et al., which detects circular polarization in the thermal-neutron capture reaction. Fermi-Dirac Quantization of Linear Systems. P. BROADBIUDGE AND C. A, HURST. Department of Mathematical Physics, The University of Adelaide, G.P.O. Box 498, Adelaide, 5001, South Australia. After discussing the Fermion analogues of classical mechanics, we show that in finite degrees of freedom, the Segal-Weinless construction of the vacuum representation is always possible. This amounts to an explicit construction of a complex structure J which extends real Euclidean space with orthogonal dynamics to a complex Hilbert space with unitary dynamics. Also, we solve the inverse problem, deducing the class of classical Hamiltonians, given the complex structure J. Geometry of Two Dimensional Tori in Phase: Projections, Sections and the Wigner Function. A. M. OZORIO DE ALMEIDA AND J. H. HANNAY. H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 lTL, England. The invariant manifolds (or “classical eigenstates”) in the phase space of bound integrable dynamical systems are known to be tori. Sections and projections of general, and special, two dimensional tori in four dimensional phase space are considered. Particular attention is paid to the families of projections accessed by linear canonical transformation since these can (in a certain sense) be considered to be different views of the same torus. The Wigner phase space representation of the corresponding semiclassical quantum eigenstate for a torus of any dimensionality is examined following the analysis of M. V. Berry (Phil. Trans. Roy. Sot. 287 (1977), 237) for one dimensional tori. In this, the value of the semiclassical Wigner function at any phase space point depends on the behaviour of the chords of the torus centred on that point. It is found that for a two dimensional torus the number of such chords is always even. The three dimensional surfaces across which the number of chords changes constitute a (double) fold catastrophe on which the function oscillates with large amplitude. On the torus manifold itself this “Wigner caustic” generally exhibits a hyperbolic umbilic singularity (possibly interspersed with elliptic regions). At special lines and points on the torus, however, higher catstrophes up to E, are generic. Legendre Transforms and r-Parricle Irreducibility in Quantum Field Theory: The Formalism for r = 1,2. ALAN COOPER, JOEL FELDMAN, AND LON ROSEN. Mathematics Department, University of British Columbia, Vancouver, British Columbia, V6T lY4, Canada. We analyze the first and second Legendre transforms f(‘) (r = 1, 2) of the generating functional G for connected Green’s functions in Euclidean boson field theories. By using Spencer’s idea of t-lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that r”’ generates r-irreducible vertex functions, r-irreducible expectations and r-field projectors; moreover, f(*) generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.

Legendre transforms and r-particle irreducibility in quantum field theory: The formalism for r = 1,2

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Page 1: Legendre transforms and r-particle irreducibility in quantum field theory: The formalism for r = 1,2

436 ABSTRACTS OF PAPERS TO APPEAR IN FUTURE ISSUES

nucleon parity-violating couplings, may be determinable from such experiments. Further, it is possible to check the experiment of Lobashov et al., which detects circular polarization in the thermal-neutron capture reaction.

Fermi-Dirac Quantization of Linear Systems. P. BROADBIUDGE AND C. A, HURST. Department of Mathematical Physics, The University of Adelaide, G.P.O. Box 498, Adelaide, 5001, South Australia.

After discussing the Fermion analogues of classical mechanics, we show that in finite degrees of freedom, the Segal-Weinless construction of the vacuum representation is always possible. This amounts to an explicit construction of a complex structure J which extends real Euclidean space with orthogonal dynamics to a complex Hilbert space with unitary dynamics. Also, we solve the inverse problem, deducing the class of classical Hamiltonians, given the complex structure J.

Geometry of Two Dimensional Tori in Phase: Projections, Sections and the Wigner Function. A. M. OZORIO DE ALMEIDA AND J. H. HANNAY. H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 lTL, England.

The invariant manifolds (or “classical eigenstates”) in the phase space of bound integrable dynamical systems are known to be tori. Sections and projections of general, and special, two dimensional tori in four dimensional phase space are considered. Particular attention is paid to the families of projections accessed by linear canonical transformation since these can (in a certain sense) be considered to be different views of the same torus. The Wigner phase space representation of the corresponding semiclassical quantum eigenstate for a torus of any dimensionality is examined following the analysis of M. V. Berry (Phil. Trans. Roy. Sot. 287 (1977), 237) for one dimensional tori. In this, the value of the semiclassical Wigner function at any phase space point depends on the behaviour of the chords of the torus centred on that point. It is found that for a two dimensional torus the number of such chords is always even. The three dimensional surfaces across which the number of chords changes constitute a (double) fold catastrophe on which the function oscillates with large amplitude. On the torus manifold itself this “Wigner caustic” generally exhibits a hyperbolic umbilic singularity (possibly interspersed with elliptic regions). At special lines and points on the torus, however, higher catstrophes up to E, are generic.

Legendre Transforms and r-Parricle Irreducibility in Quantum Field Theory: The Formalism for r = 1,2. ALAN COOPER, JOEL FELDMAN, AND LON ROSEN. Mathematics Department, University of British Columbia, Vancouver, British Columbia, V6T lY4, Canada.

We analyze the first and second Legendre transforms f(‘) (r = 1, 2) of the generating functional G for connected Green’s functions in Euclidean boson field theories. By using Spencer’s idea of t-lines we define and prove irreducibility properties independently of perturbation theory. In particular we prove that r”’ generates r-irreducible vertex functions, r-irreducible expectations and r-field projectors; moreover, f(*) generates (generalized) Bethe-Salpeter kernels with 2-cluster-irreducibility properties.