Dirac Matrices and Feynman’s Rest of the Universe

Kim, Young S.; Noz, Marilyn E.

doi:10.3390/sym4040626

Cited by 8 publications

(7 citation statements)

References 28 publications

(66 reference statements)

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“…Refs. [29][30][31][32][33]. As a consequence, it may be skipped by those readers already familiar with the use of symplectic groups in quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

2022

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We construct the four-mode squeezed states and study their physical properties. These states describe two linearly-coupled quantum scalar fields, which makes them physically relevant in various contexts such as cosmology. They are shown to generalise the usual two-mode squeezed states of single-field systems, with additional transfers of quanta between the fields. To build them in the Fock space, we use the symplectic structure of the phase space. For this reason, we first present a pedagogical analysis of the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$ Sp ( 4 , R ) and its Lie algebra, from which we construct the four-mode squeezed states and discuss their structure. We also study the reduced single-field system obtained by tracing out one of the two fields. This procedure being easier in the phase space, it motivates the use of the Wigner function which we introduce as an alternative description of the state. It allows us to discuss environmental effects in the case of linear interactions. In particular, we find that there is always a range of interaction coupling for which decoherence occurs without substantially affecting the power spectra (hence the observables) of the system.

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“…Refs. [29][30][31][32][33]. As a consequence, it may be skipped by those readers already familiar with the use of symplectic groups in quantum mechanics.…”

Section: Introductionmentioning

confidence: 99%

Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

2022

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“…The chiral operator, denoted by γ 5 (or γ 5 in old papers), appears many times in the literature. It is defined by γ 5 = iω 1,3 , i.e., γ 5 = iγ 0 γ 1 γ 2 γ 3 (see [21,27,30,40,44,46,47,[49][50][51][52][53][54]), or by γ 5 = i 3 γ 0 γ 1 γ 2 γ 3 (see [33,54,55]). From Corollary 3, γ 5 anticommutes with any of the other gamma matrices, and from Corollary 4, (γ 5 ) 2 = 1.…”

Section: Direct and Indirect Symmetry; Chiralitymentioning

confidence: 99%

Chirality of Dirac Spinors Revisited

Petitjean

2020

Symmetry

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We emphasize the differences between the chirality concept applied to relativistic fermions and the ususal chirality concept in Euclidean spaces. We introduce the gamma groups and we use them to classify as direct or indirect the symmetry operators encountered in the context of Dirac algebra. Then we show how a recent general mathematical definition of chirality unifies the chirality concepts and resolve conflicting conclusions about symmetry operators, and particularly about the so-called chirality operator. The proofs are based on group theory rather than on Clifford algebras. The results are independent on the representations of Dirac gamma matrices, and stand for higher dimensional ones.

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“…( 39) can serve as the four-dimensional symplectic group Sp(4). This group allows us to study squeezed or entangled states in terms of the four-dimensional phase space consisting of two position and two momentum variables [15,40,41].…”

Section: Dirac's Entangled Oscillatorsmentioning

confidence: 99%

Entangled Harmonic Oscillators and Space-Time Entanglement

2016

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Abstract:The mathematical basis for the Gaussian entanglement is discussed in detail, as well as its implications in the internal space-time structure of relativistic extended particles. It is shown that the Gaussian entanglement shares the same set of mathematical formulas with the harmonic oscillator in the Lorentz-covariant world. It is thus possible to transfer the concept of entanglement to the Lorentz-covariant picture of the bound state, which requires both space and time separations between two constituent particles. These space and time variables become entangled as the bound state moves with a relativistic speed. It is shown also that our inability to measure the time-separation variable leads to an entanglement entropy together with a rise in the temperature of the bound state. As was noted by Paul A. M. Dirac in 1963, the system of two oscillators contains the symmetries of the O(3, 2) de Sitter group containing two O(3, 1) Lorentz groups as its subgroups. Dirac noted also that the system contains the symmetry of the Sp(4) group, which serves as the basic language for two-mode squeezed states. Since the Sp(4) symmetry contains both rotations and squeezes, one interesting case is the combination of rotation and squeeze, resulting in a shear. While the current literature is mostly on the entanglement based on squeeze along the normal coordinates, the shear transformation is an interesting future possibility. The mathematical issues on this problem are clarified.

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Dirac Matrices and Feynman’s Rest of the Universe

Cited by 8 publications

References 28 publications

Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

Four-mode squeezed states: two-field quantum systems and the symplectic group $$\mathrm {Sp}(4,{\mathbb {R}})$$

Chirality of Dirac Spinors Revisited

Entangled Harmonic Oscillators and Space-Time Entanglement

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