The recently discovered narrow peaks (theψ-particles) in e⁺e⁻ system at 3.105 and 3.695 GeV are interpreted as hadrons in a broken SU₄ symmetry scheme. A new additional additive quantum number, parachargeZ, is combined with the usual SU₃ quantum numbers in the group SU₄. Theψ (3.1) is assigned to a near ideally mixed15⊕1 multiplet of vector mesons (containing theρ) as theI=Y=0, charge conjugationC=−combination ofZ=±1.members. Theψ (3.7) is assigned correspondingly to another mixed15⊕1 multiplet containing theρ′ (1600). The hadronic electromagnetic interactions are modified by the addition of (non-minimal) anomalous pieces that can changeZ. The decays of theψ-particles are discussed. New enlarged SU₄ multiplets of other hadrons are proposed. Tests of our scheme are put forward. The most crucial test will be the observation of two rather broad resonances in e⁺ e⁻ collisions with masses around 4.2 GeV and 5.1 GeV. Another prediction is the presence of energetic photons in the decays of theψ-particles. Important results concerning the recently observed phenomena in the process e⁺e⁻→hadrons follow in this scheme.

A universal theory of weak interaction is constructed by exploiting an analogy inherent between the four leptons and the four quarks of the paracharge scheme proposed recently to deal with theψ-particles. The leptons (ν_e,ν_μ,e_L,μ_L) are assigned to the representation (1/2, 1/2) and the quarks (p, n_W)_L and (χ,λ_W)_L to the representations (1/2, 0) and (0, 1/2), respectively, of the groupO₄ (L stands for the left-handed projections and W for the Cabibbo rotated orthogonal combinations ofn andλ). Universality is ensured by embedding the above (weak)O₄ into thesimple groupO₅ and gauging the latter. In the final effective weak interaction, besides the conventionalV-A charged-current part, a (V-A)neutral current interaction (consistent with the present data) is naturally present. The neutral current has a$$\bar \nu _\mu \nu _\mu $$ term but no$$\bar \nu _e \nu _e $$ term, thus providing a crucial test of the theory.

A systematic semiquantitative account of all aspects of the strong and electromagnetic interactions of all the newly discovered hadronic states (theψ’s, theχ’s, etc.) is presented within the framework of the paracharge scheme. Extensions of ideas familiar from the SU₃ classification scheme to SU₄ are shown to provide an understanding of the new states seen in the decays ofψ (3.1) andψ′ (3.7), including their masses and gross decay characteristics. The decays ofψ (3.1) andψ′ (3.7) themselves are studied in some detail. Since these are of electromagnetic origin in the scheme, their electromagnetic mixing with the resonance at 4.15 GeV (theP-state of the scheme) is important. Once this is taken into account, the resulting picture is in excellent agreement with available data.

We show that for a fermion in a bounded background potential in a finite box, eigenvalues of the total charge are independent of whether the potential is solitonic and depend only on the boundary condition: half-odd integral or integral for charge conjugation (C) invariant boundary conditions and an arbitrary fraction forC non-invariant boundary conditions. Fractional fermion numbers for infinite space Jackiw-Rebbi and Goldstone-Wilczek Hamiltonians are reproduced in finite space by using boundary conditions different from the periodic ones of Rajaraman and Bell.

We propose an analytic perturbative approach for the determination of the Feigenbaum-Cvitanović function and the universal parameterα occurring in the Feigenbaum scenario of period doubling for approach to chaotic behaviour. We apply the method to the caseZ=2 whereZ is the order of the unique local maximum of the nonlinear map. Our third order approximation givesα=2.5000 as compared to “exact” numerical valueα=2.5029 ... We also obtain a reasonably accurate value of the Feigenbaum-Cvitanović function.

We consider a fermion of chargee confined to a spherical bag with a Dirac monopole of strengthg at its centre. We find that the boundary conditions making the lowest angular momentum hamiltonian self-adjoint are characterized by a unitary matrixU, and the corresponding vacuum charge has a fractional part 2|eg|α/π where detU = -exp (2iα). Boundary conditions for conservation of helicity,CP, CT andPT are displayed. We demonstrate the possibility of a fractionally charged dyon whose interaction with a fermion conserves helicity. We also show thatthe simultaneous validity of helicity, CP, CT and PT requires integer vacuum charge.

Belavin, Zamlodochikov and Polyakov have recently proposed a class of conformally invariant field theories in two dimension with exactly determined rational critical indices. We establish a tentative identification of a subset of these theories in terms of the O(n) model and theq-state Potts model in 2-dimensions for appropriaten andq.

We show that the classical Nambu-Goto string inD dimensions admits Poincaré invariance ind dimensions (d⩽D) if (i)d − 2 of the transverse co-ordinatesxⁱ are periodic and the rest quasi-periodic involving a real orthogonal matrix with (D − d) (D − d − 1)/2 free parameters, or if (ii)d − 2 ofxⁱ obey Neumann and the rest obey a boundary condition involvingN free parameters, whereN=(D − d)²/2 ifD − d is even, andN=[(D − d)² − 1]/2 ifD − d is odd.

We discuss the split between system and measuring apparatus, i.e. non-unitary nature, of the conventional quantum mechanics to motivate a causal unitary description of nature. We then describe causal quantum mechanics of de Broglie and Bohm. We conclude by presenting a version of recently proposed causal quantum mechanics which treats position and momentum variable symmetrically.