Group Theoretical Approach to Pseudo-Hermitian Quantum Mechanics with Lorentz Covariance and c → ∞ Limit

Abstract

We present the formulation of a version of Lorentz covariant quantum mechanics based on a group theoretical construction from a Heisenberg–Weyl symmetry with position and momentum operators transforming as Minkowski four-vectors. The basic representation is identified as a coherent state representation, essentially an irreducible component of the regular representation, with the matching representation of an extension of the group C*-algebra giving the algebra of observables. The key feature is that it is not unitary but pseudo-unitary, exactly in the same sense as the Minkowski spacetime representation. The language of pseudo-Hermitian quantum mechanics is adopted for a clear illustration of the aspect, with a metric operator obtained as really the manifestation of the Minkowski metric on the space of the state vectors. Explicit wavefunction description is given without any restriction of the variable domains, yet with a finite integral inner product. The associated covariant harmonic oscillator Fock state basis has all the standard properties in exact analog to those of a harmonic oscillator with Euclidean position and momentum operators. Galilean limit and the classical limit are retrieved rigorously through appropriate symmetry contractions of the algebra and its representation, including the dynamics described through the symmetry of the phase space.

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“…Such representation theory properly extended gives essentially the full theory and its approximate versions as contraction limits [5]. We have put together under a single framework the group theoretical formulation of a Lorentz invariant quantum mechanics and all its approximations as the 'nonrelativistic' and/ or classical limits [16]. The theory has a notion of noncommutative Minkowski spacetime and a Minkowski metric operator on the corresponding Hilbert space.…”

“…Otani has been leading the muon acceleration program for the muon g-2/EDM experiment at J-PARC since 2013, after receiving his PhD in physics for the measurement of neutrino oscillation in the T2K long baseline experiment. Yangping Shen (谌阳平), Department of Nuclear Physics, China Institute of Atomic Energy "For his contribution in shedding new light on the critical reaction, 12 C(α, γ) 16 O, in nuclear astrophysics" 12 C(α, γ) 16 O is regarded as the "Holy Grail" reaction in the nuclear astrophysical research, which affects the synthesis of the elements up to iron and governs stellar evolution, including the formation of black holes or neutron stars as massive stars exhaust their energy.…”

“…Shen's works mainly focus on the direct and indirect measurement of this "Holy Grail" reaction. He has developed an indirect technique based on the independent ( 11 B, 7 Li) transfer reaction with less breakup effect and has experimentally determined the external-capture contribution in 12 C(α, γ) 16 O for the first time. This work results in a significant increase of the total S factor, which is now in good agreement with the value obtained by reproducing supernova nucleosynthesis calculations with solar-system abundances.…”

Special Relativity and Its Newtonian Limit from a Group Theoretical Perspective

“…The symmetry for the 'nonrelativistic' quantum mechanics is essentially a U(1) central extension [6,7] of the Galilean symmetry, or the one with the Newtonian time translation taken out-we called that H R (3) as the Heisenberg-Weyl symmetry with three noncommuting X-P pairs supplemented by the SO(3) [3,5]. For the 'relativistic' case, we found it necessary to go beyond the Poincaré symmetry to the larger H R (1, 3) [4,5]. The last reference also addresses nonzero spin and composite systems.…”

“…Secondly, a particular way to look at a theory, even if not in any sense superior to the other ways, may provide a specific channel to go to theories beyond. The authors' attention on the subject matter is closely connected to our recent studies essentially on the exact parallel constructions for the theories of quantum mechanics, including retrieving of the 'nonrelativistic' from the 'relativistic' as well as classical from quantum [3,4]. The Lorentz covariant theory of quantum mechanics resulted is new, with a quantum notion of Minkowski metric.…”