Functional Quantum Theory of Free Relativistic Fermi Fields

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.

scholarly journals On alienation of two functional equations of quadratic type

2021 ◽  
Author(s):  
Roman Ger
Keyword(s):  
Functional Equation ◽  
Functional Equations ◽  
Quadratic Functional Equation ◽  
Quadratic Functional ◽  
Real Numbers ◽  
Alpha Beta ◽  
Beta 2 ◽  
Special Case

Abstract  We deal with an alienation problem for an Euler–Lagrange type functional equation $$\begin{aligned} f(\alpha x + \beta y) + f(\alpha x - \beta y) = 2\alpha ^2f(x) + 2\beta ^2f(y) \end{aligned}$$ f ( α x + β y ) + f ( α x - β y ) = 2 α 2 f ( x ) + 2 β 2 f ( y ) assumed for fixed nonzero real numbers $$\alpha ,\beta ,\, 1 \ne \alpha ^2 \ne \beta ^2$$ α , β , 1 ≠ α 2 ≠ β 2 , and the classic quadratic functional equation $$\begin{aligned} g(x+y) + g(x-y) = 2g(x) + 2g(y). \end{aligned}$$ g ( x + y ) + g ( x - y ) = 2 g ( x ) + 2 g ( y ) . We were inspired by papers of Kim et al. (Abstract and applied analysis, vol. 2013, Hindawi Publishing Corporation, 2013) and Gordji and Khodaei (Abstract and applied analysis, vol. 2009, Hindawi Publishing Corporation, 2009), where the special case $$g = \gamma f$$ g = γ f was examined.

scholarly journals Effective Hamiltonians and Wavefunctions for Electrons in Deformed Crystals

1983 ◽  
Vol 36 (3) ◽  
pp. 321 ◽  
Author(s):  
RA Brown
Keyword(s):  
Hilbert Space ◽  
Probability Density ◽  
Physical Interpretation ◽  
Wannier Function ◽  
Deformation Potential ◽  
Effective Hamiltonian ◽  
Wannier Functions ◽  
Modulating Functions ◽  
Strain Dependent ◽  
Effective Hamiltonians

An effective Hamiltonian for electrons in in homogeneously deformed crystals is derived by expanding the wavefunction in terms of Wannier functions of the homogeneously deformed crystal. The physical interpretation of the modulating functions which determine the amplitude of each Wannier function in the expansion, and which are governed by the effective Hamiltonian, is investigated. This leads to strain-dependent expressions for the probability density and current, averaged over the fluctuations within each unit cell. The operators which represent, in the Hilbert space of the . modulating functions, similarly averaged physical observables are introduced and explicit straindependent expressions for the velocity and momentum operators are obtained. Applications of the theory are foreshadowed and its relationship to previous deformation-potential theories is examined.

scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

On Operator Algebras and Invariant Subspaces

1969 ◽  
Vol 21 ◽  
pp. 1178-1181 ◽  
Author(s):  
Chandler Davis ◽  
Heydar Radjavi ◽  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Elementary Proof ◽  
Operator Algebras ◽  
Equivalent Form ◽  
Invariant Subspaces ◽  
Complex Hilbert Space ◽  
Closed Algebra ◽  
Weakly Closed ◽  
Special Case

If is a collection of operators on the complex Hilbert space , then the lattice of all subspaces of which are invariant under every operator in is denoted by Lat . An algebra of operators on is defined (3; 4) to be reflexive if for every operator B on the inclusion Lat ⊆ Lat B implies .Arveson (1) has proved the following theorem. (The abbreviation “m.a.s.a.” stands for “maximal abelian self-adjoint algebra”.)ARVESON's THEOREM. Ifis a weakly closed algebra which contains an m.a.s.a.y and if Lat, then is the algebra of all operators on .A generalization of Arveson's Theorem was given in (3). Another generalization is Theorem 2 below, an equivalent form of which is Corollary 3. This theorem was motivated by the following very elementary proof of a special case of Arveson's Theorem.

scholarly journals Space-time from Collapse of the Wave-function

2019 ◽  
Vol 74 (2) ◽  
pp. 147-152 ◽  
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.

On the Calculation of Nonlinear Spinor Field Functionals

1971 ◽  
Vol 26 (4) ◽  
pp. 623-630 ◽  
Author(s):  
H Stumpf
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Quantum Theory ◽  
Spinor Field ◽  
Functional Equations ◽  
High Energy ◽  
Quantum Field ◽  
Nonlinear Spinor ◽  
Physical Problems ◽  
Physical Information

Abstract Dynamics of quantum field theory can be formulated by functional equations. To develop a complete functional quantum theory one has to describe the physical information by functional operations only. Such operations have been defined in preceding papers. To apply these operations to physical problems, the corresponding functionals have to be known. Therefore in this paper calculational procedures for functionals are discussed. As high energy phenomena are of interest, the calculational procedures are given for spinor field functionals. Especially a method for the calculation of stationary and Fermion-Fermion scattering functionals is proposed.

scholarly journals Quantum Theory in Pseudo-Hilbert Space. II

1959 ◽  
Vol 21 (5) ◽  
pp. 727-730 ◽  
Author(s):  
Gaku Konisi ◽  
Takesi Ogimoto
Keyword(s):  
Hilbert Space ◽  
Quantum Theory

On the Quantum Theory of the Anharmonic Oscillator in Functional Space

1969 ◽  
Vol 24 (2) ◽  
pp. 188-197 ◽  
Author(s):  
H. Stumpf
Keyword(s):  
Field Theory ◽  
Quantum Theory ◽  
Functional Equations ◽  
Scalar Product ◽  
Anharmonic Oscillator ◽  
Functional Space ◽  
Solution Procedure ◽  
Field Theories ◽  
Quantum Field ◽  
Product Definition

Dynamics of quantum field theory can be formulated by functional equations. For strong inter­action nonperturbative solutions of these functional equations are required. For the investigation of solution procedures the model of an anharmonic oscillator is used, because of its structural equi­valence with dressed one- and two-particel states of field theory. To perform a variational solution procedure a scalar product for the state functionals is introduced and its existence is proven. The scalar product definition admits a mapping of the physical Hilbert space on the functional space. Therefore a “functional” quantum theory seems to be possible. The whole procedure can be transferred to relativistic invariant field theories, provided these theories are regularized to give finite results at all.

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