Second quantization for systems with a constant number of particles in the Dirac notation

1977 ◽  
Vol 11 (3) ◽  
pp. 505-523 ◽  
Author(s):  
Jaroslav Koutecký ◽  
Alexandre Laforgue
Keyword(s):  
Constant Number ◽  
Second Quantization ◽  
Number Of Particles

scholarly journals Entanglement in the second quantization formalism

Open Physics ◽  
2003 ◽  
Vol 1 (2) ◽  
Author(s):  
Vlatko Vedral
Keyword(s):  
Degrees Of Freedom ◽  
Bose Condensation ◽  
Unruh Effect ◽  
Correlated Electrons ◽  
Second Quantization ◽  
Bipartite Entanglement ◽  
Entanglement Of Formation ◽  
Quantization Formalism ◽  
Internal Degrees Of Freedom ◽  
Number Of Particles

AbstractWe study properties of entangled systems in the (mainly non-relativistic) second quantization formalism. This is then applied to interacting and non-interacting bosons and fermions and the differences between the two are discussed. We present a general formalism to show how entanglement changes with the change of modes of the system. This is illustrated with examples such as the Bose condensation and the Unruh effect. It is then shown that a non-interacting collection of fermions at zero temperature can be entangled in spin, providing that their distances do not exceed the inverse Fermi wavenumber. Beyond this distance all bipartite entanglement vanishes, although classical correlations still persist. We compute the entanglement of formation as well as the mutual information for two spin-correlated electrons as a function of their distance. The analogous, non-interacting collection of bosons displays no entanglement in the internal degrees of freedom. We show how to generalize our analysis of the entanglement in the internal degrees of freedom to an arbitrary number of particles.

scholarly journals Quantum Techniques for Reaction Networks

2018 ◽  
Vol 2018 ◽  
pp. 1-9 ◽  
Author(s):  
John C. Baez
Keyword(s):  
Master Equation ◽  
Rate Equation ◽  
Quantum Physics ◽  
Time Derivative ◽  
Reaction Network ◽  
Reaction Networks ◽  
Expected Number ◽  
Second Quantization ◽  
Stochastic Time ◽  
Number Of Particles

Reaction networks are a general formalism for describing collections of classical entities interacting in a random way. While reaction networks are mainly studied by chemists, they are equivalent to Petri nets, which are used for similar purposes in computer science and biology. As noted by Doi and others, techniques from quantum physics, such as second quantization, can be adapted to apply to such systems. Here we use these techniques to study how the “master equation” describing stochastic time evolution for a reaction network is related to the “rate equation” describing the deterministic evolution of the expected number of particles of each species in the large-number limit. We show that the relation is especially strong when a solution of master equation is a “coherent state”, meaning that the numbers of entities of each kind are described by independent Poisson distributions. Remarkably, in this case the rate equation and master equation give the exact same formula for the time derivative of the expected number of particles of each species.

The Nature of the Photon and Quantum Optics

2018 ◽  
pp. 4-13 ◽  
Author(s):  
Boris A. Veklenko
Keyword(s):  
Electromagnetic Field ◽  
Quantum Electrodynamics ◽  
Quantitative Description ◽  
Wave Functions ◽  
Second Quantization ◽  
Interaction Processes ◽  
Quantized Field ◽  
Transverse Electromagnetic ◽  
Set Up ◽  
Number Of Particles

In a concise, but accessible for the first acquaintance form the procedure for the quantization of linear oscillator is set out. By analogy with this procedure the procedure of quantization (second quantization) of classical Maxwell’s electrodynamics is set up. The physical sense of the wave functions arguments of transverse electromagnetic field and its Fourier transformation is set up. One pay attention as for quantum coherent (almost a classic) states of the electromagnetic field and for photonics Fock states. Attention is drawn to the fact of absence the power of the universal content of such concepts as field amplitude, phase and number of particles (photons), which are used by experimenter’s to describe the states of a quantized field. The semi quantitative description the interaction processes of a quantum electromagnetic field with substance is set up. Specified situations are shown in which the discrepancy between the predictions of classical and quantum electrodynamics is noticeable at the macroscopic level.

A note on relativistic second quantization

1947 ◽  
Vol 43 (2) ◽  
pp. 183-195 ◽  
Author(s):  
T. S. Chang
Keyword(s):  
Scalar Product ◽  
Wave Equations ◽  
Relativistic Wave ◽  
Second Quantization ◽  
Time Coordinate ◽  
Creation And Annihilation Operators ◽  
Scalar Particles ◽  
Know How ◽  
Free Particles ◽  
Number Of Particles

The relativistic second quantization of free bosons is extended to fermions. To know how relativistic creation and annihilation operators operate on bra and ket vectors, it is necessary to have a relativistic scalar product which in the case of free particles can be constructed. For particles interacting with each other or capable of emitting and absorbing particles of the same kind, it is pointed out that adequate wave equations for an indefinite number of particles each taking a separate time coordinate have not yet been found and so no scalar product can be found. Thus creation and annihilation operators can be defined only by their operation on a bra vector or a ket vector but not by both. With the help of these, the relativistic wave equations referred to above are proposed and their consistency conditions studied. For the particular case of scalar particles, an illustration is given that such a theory may admit certain expressions as probabilities.

scholarly journals Vacuum and Spacetime Signature in the Theory of Superalgebraic Spinors

Universe ◽  
2019 ◽  
Vol 5 (7) ◽  
pp. 162 ◽  
Author(s):  
Vadim Monakhov
Keyword(s):  
Degrees Of Freedom ◽  
Vacuum State ◽  
Second Quantization ◽  
New Formalism ◽  
Gauge Transformations ◽  
Dimensional Spacetime ◽  
Internal Degrees Of Freedom ◽  
Natural Way ◽  
Number Of Particles

A new formalism involving spinors in theories of spacetime and vacuum is presented. It is based on a superalgebraic formulation of the theory of algebraic spinors. New algebraic structures playing role of Dirac matrices are constructed on the basis of Grassmann variables, which we call gamma operators. Various field theory constructions are defined with use of these structures. We derive formulas for the vacuum state vector. Five operator analogs of five Dirac gamma matrices exist in the superalgebraic approach as well as two additional operator analogs of gamma matrices, which are absent in the theory of Dirac spinors. We prove that there is a relationship between gamma operators and the most important physical operators of the second quantization method: number of particles, energy–momentum and electric charge operators. In addition to them, a series of similar operators are constructed from the creation and annihilation operators, which are Lorentz-invariant analogs of Dirac matrices. However, their physical meaning is not yet clear. We prove that the condition for the existence of spinor vacuum imposes restrictions on possible variants of the signature of the four-dimensional spacetime. It can only be (1, − 1 , − 1 , − 1 ), and there are two additional axes corresponding to the inner space of the spinor, with a signature ( − 1 , − 1 ). Developed mathematical formalism allows one to obtain the second quantization operators in a natural way. Gauge transformations arise due to existence of internal degrees of freedom of superalgebraic spinors. These degrees of freedom lead to existence of nontrivial affine connections. Proposed approach opens perspectives for constructing a theory in which the properties of spacetime have the same algebraic nature as the momentum, electromagnetic field and other quantum fields.

scholarly journals THERMODYNAMICS OF PSEUDO-HERMITIAN SYSTEMS IN EQUILIBRIUM

2007 ◽  
Vol 22 (15) ◽  
pp. 1075-1084 ◽  
Author(s):  
VÍT JAKUBSKÝ
Keyword(s):  
Thermodynamic Properties ◽  
Physical Interpretation ◽  
Explicit Knowledge ◽  
Positive Definite ◽  
Explicit Calculation ◽  
Constant Number ◽  
Metric Operator ◽  
Number Of Particles

In the study of pseudo(quasi)-hermitian operators, the key role is played by the positive-definite metric operator. It enables physical interpretation of the considered systems. In this paper, we study the pseudo-hermitian systems with constant number of particles in equilibrium. We show that the explicit knowledge of the metric operator is not essential for the study of thermodynamic properties of the system. We introduce a simple example where the physically relevant quantities are derived without explicit calculation of either metric operator or spectrum of the Hamiltonian.

Stereo Electron Microscopy Applied to High Resolution Freeze-Etch Replicas

1970 ◽  
Vol 28 ◽  
pp. 306-307
Author(s):  
L. Andrew Staehelin
Keyword(s):  
Electron Microscopy ◽  
Plastic Deformation ◽  
High Resolution ◽  
Smooth Surfaces ◽  
Small Particles ◽  
Membrane Surfaces ◽  
Wide Acceptance ◽  
New Information ◽  
Number Of Particles ◽  
Small Holes

Freeze-etched membranes usually appear as relatively smooth surfaces covered with numerous small particles and a few small holes (Fig. 1). In 1966 Branton (1“) suggested that these surfaces represent split inner mem¬brane faces and not true external membrane surfaces. His theory has now gained wide acceptance partly due to new information obtained from double replicas of freeze-cleaved specimens (2,3) and from freeze-etch experi¬ments with surface labeled membranes (4). While theses studies have fur¬ther substantiated the basic idea of membrane splitting and have shown clearly which membrane faces are complementary to each other, they have left the question open, why the replicated membrane faces usually exhibit con¬siderably fewer holes than particles. According to Branton's theory the number of holes should on the average equal the number of particles. The absence of these holes can be explained in either of two ways: a) it is possible that no holes are formed during the cleaving process e.g. due to plastic deformation (5); b) holes may arise during the cleaving process but remain undetected because of inadequate replication and microscope techniques.

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