Remarks on quantum statistics

Marcinek, Władysław

doi:10.1063/1.57122

Cited by 3 publications

(2 citation statements)

References 45 publications

(67 reference statements)

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“…The first step concerns the transformation of the initial particles under interaction into composite systems consisting of quasi-particles and quanta. Such systems represent possible results of interactions [17,20,22,21]. In the second step, we describe the algebra of realizations of quantum possibilities.…”

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Quantum-classical interactions and galois type extensions

Marcinek

2003

Banach Center Publications

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Abstract. An algebraic model for the relation between a certain classical particle system and the quantum environment is proposed. The quantum environment is described by the category of possible quantum states. The initial particle system is represented by an associative algebra in the category of states. The key new observation is that particle interactions with the quantum environment can be described in terms of Hopf-Galois theory. This opens up a possibility to use quantum groups in our model of particle interactions.

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“…The problem how such 'decisions' are made was solved in [19] with the help of quantum commutativity and generalized Pauli exclusion principle. Our approach is based on the previously developed concept of particle systems with generalized statistics and quantum symmetries [15,17,20,22,21,23].…”

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Quantum-classical interactions and galois type extensions

Marcinek

2003

Banach Center Publications

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Elementary State

Siegel¹,

Zachos²,

Spallucci³

et al. 2004

Concise Encyclopedia of Supersymmetry

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A noncritical string in six dimensions with ð0; 1Þ spacetime supersymmetry [1] which is dual to the heterotic string with an E 8 instanton of zero size [2], and which in F-theory corresponds to a particular P 1 shrinking to zero size [3]. The ð0; 1Þ noncritical string can be viewed as an M5 brane wrapped around 1 2 K 3. The study of the BPS states of E-strings was made in [4].The characters of the n-string bound state are captured by N ¼ 4 U ðnÞ topological Yang-Mills theory on 1 2 K 3, and this relation can also be used to shed light on the properties of N ¼ 4 topological Yang-Mills theories on manifolds with b þ 2 ¼ 1 [5]. In particular the E-string partition functions, which can be computed using local mirror symmetry on a Calabi-Yau threefold, give the Euler characteristics of the Yang-Mills instanton moduli space on 1 2 K 3. The partition functions are determined by a gap condition combined with a simple recurrence relation which has its origins in a holomorphic anomaly that has been conjectured to exist for N ¼ 4 topological Yang-Mills on manifolds with b þ 2 ¼ 1 and is also related to the holomorphic anomaly for higher genus topological strings on Calabi-Yau threefolds [5]. The case with n < 9 is important for counting BPS states for fivedimensional versions of E-strings [6]. Bibliography Steven DuplijECTOPLASM A superspace approach to component expansion of supergravity actions, based on recognizing component actions as differential forms on bosonic subspaces of the full superspace [1]. The basic idea is that any integrand is a differential form, and any integral over a subspace is independent of the choice of surface only if that form is closed. The most familiar example is a conserved charge: The integral of the time component of a current over space is conserved if the current is divergenceless; it is completely independent of spacelike hypersurface (locally conserved) only if the current is so. The current is then actually a ðD À 1Þ-form in a Ddimensional space, and the current conservation law is actually the statement that it is curl free, which does not rely on a metric:In superspace this condition is solved after converting the curved indices to flat: For any p-form J , dJ ¼ 0 ) 1 p r ½A J BC :::DÞ ¼ 1 2 T ½ABj E J EjC :::DÞ in terms of the covariant derivative r A and torsion T AB C . Spinor indices carry a lower engineering dimension than vector ones, so the constraints with more spinor indices are solved first. Many of the higherdimension components then follow by using the constant part of the torsion. Similar remarks apply to solving more general constraints, such as for the field strengths of physical forms like the vector and tensor multiplets. In fact, the D-form that describes component Lagrangians can be derived inductively in p for general p-forms, since they satisfy generic sets of constraints, independently of what type of object they are being used to describe: For any system where the gauge parameters are p-forms, the gauge fields are ðp þ 1Þ-forms, the field strengths are ðp þ 2...

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On crossed product of algebras

Borowiec

Marcinek

2000

Journal of Mathematical Physics

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The concept of a crossed tensor product of algebras is studied from a few points of views. Some related constructions are considered. Crossed enveloping algebras and their representations are discussed. Applications to the noncommutative geometry and particle systems with generalized statistics are indicated.Comment: 20 pages, latex2e (latexsym), no figures, accepted for publication in J. Math. Phys. 41 no10 (2000

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Remarks on quantum statistics

Cited by 3 publications

References 45 publications

Quantum-classical interactions and galois type extensions

Quantum-classical interactions and galois type extensions

Elementary State

On crossed product of algebras

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