Integration in Hilbert Space and Quantum Theory

Functional Quantum Theory of Free Relativistic Fermi Fields

Zeitschrift für Naturforschung A ◽

10.1515/zna-1970-0501 ◽

1970 ◽

Vol 25 (5) ◽

pp. 575-586

Author(s):

H. Stumpf

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

Physical Interpretation ◽

Functional Equations ◽

Dirac Field ◽

Functional Hilbert Space ◽

Free Fermi ◽

Special Case ◽

Relativistic Fermi

Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.

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Space-time from Collapse of the Wave-function

Zeitschrift für Naturforschung A ◽

10.1515/zna-2018-0477 ◽

2019 ◽

Vol 74 (2) ◽

pp. 147-152 ◽

Cited By ~ 3

Author(s):

Tejinder P. Singh

Keyword(s):

Hilbert Space ◽

Wave Function ◽

Quantum Theory ◽

Minkowski Space ◽

Time Scales ◽

Quantum Interference ◽

Quantum Dynamics ◽

Space Time ◽

Minkowski Space Time ◽

Macroscopic Objects

AbstractWe propose that space-time results from collapse of the wave function of macroscopic objects, in quantum dynamics. We first argue that there ought to exist a formulation of quantum theory which does not refer to classical time. We then propose such a formulation by invoking an operator Minkowski space-time on the Hilbert space. We suggest relativistic spontaneous localisation as the mechanism for recovering classical space-time from the underlying theory. Quantum interference in time could be one possible signature for operator time, and in fact may have been already observed in the laboratory, on attosecond time scales. A possible prediction of our work seems to be that interference in time will not be seen for ‘time slit’ separations significantly larger than 100 attosecond, if the ideas of operator time and relativistic spontaneous localisation are correct.

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Quantum Theory in Pseudo-Hilbert Space. II

Progress of Theoretical Physics ◽

10.1143/ptp.21.727 ◽

1959 ◽

Vol 21 (5) ◽

pp. 727-730 ◽

Cited By ~ 1

Author(s):

Gaku Konisi ◽

Takesi Ogimoto

Keyword(s):

Hilbert Space ◽

Quantum Theory

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QUANTUM BACKREACTION (CASIMIR) EFFECT WITHOUT INFINITIES: ALGEBRAIC ANALYSIS

International Journal of Modern Physics Conference Series ◽

10.1142/s2010194512007490 ◽

2012 ◽

Vol 14 ◽

pp. 376-382

Author(s):

ANDRZEJ HERDEGEN

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

Quantum System ◽

Casimir Effect ◽

Quantum Field ◽

Expectation Value ◽

Physical Systems ◽

Algebra Of Observables ◽

General Terms ◽

External Conditions

Casimir effect, in most general terms, is the backreaction of a quantum system responding to an adiabatic change of external conditions. This backreaction is expected to be quantitatively measured by a change in the expectation value of a certain energy observable of the system. However, for this concept to be applicable, the system has to retain its identity in the process. Most prevailing tendencies in the analysis of the effect seem to ignore this question. In general, a quantum theory is defined by an algebra of observables, whose representations by operators in a Hilbert space define concrete physical systems described by the theory. A quantum system retains its identity if both the algebra as well as its representation do not change. We discuss the resulting restrictions for admissible models of changing external conditions. These ideas are applied to quantum field models. No infinities arise, if the algebraic demands are respected.

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Correlations in a General Theory of Quantum Measurement

Open Systems & Information Dynamics ◽

10.1007/s11080-007-9067-x ◽

2007 ◽

Vol 14 (04) ◽

pp. 445-458 ◽

Cited By ~ 3

Author(s):

Hanna Podsędkowska

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

General Theory ◽

Quantum Measurement ◽

Von Neumann Algebra ◽

Classical Case ◽

Von Neumann ◽

Full Algebra ◽

Algebraic Formalism ◽

Algebra Of Operators

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states — by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.

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