From Classical to Quantum Fields. Free Fields

2021 ◽  
pp. 220-236
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Hilbert Space ◽  
Scalar Field ◽  
Harmonic Oscillator ◽  
Path Integral ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Grassmann Algebra ◽  
Commutation Relations ◽  
Spinor Fields ◽  
Covariant Gauge

We apply the canonical and the path integral quantisation methods to scalar, spinor and vector fields. The scalar field is a generalisation to an infinite number of degrees of freedom of the single harmonic oscillator we studied in Chapter 9. For the spinor fields we show the need for anti-commutation relations and introduce the corresponding Grassmann algebra. The rules of Fermi statistics follow from these anti-commutation relations. The canonical quantisation method applied to the Maxwell field in a Lorentz covariant gauge requires the introduction of negative metric states in the Hilbert space. The power of the path integral quantisation is already manifest. In each case we expand the fields in creation and annihilation operators.

scholarly journals Scalar field effective action for the spontaneous symmetry breaking in gravity

2019 ◽  
Vol 34 (20) ◽  
pp. 1950156
Author(s):  
Amin Akhavan
Keyword(s):  
Vector Field ◽  
Scalar Field ◽  
Symmetry Breaking ◽  
Path Integral ◽  
Effective Action ◽  
Degrees Of Freedom ◽  
Special Kind ◽  
Canonical Momentum ◽  
Massive Vector ◽  
The One

In this paper, we obtain the quantum effective action for a scalar field which has been used for a special kind of symmetry breaking in gravity. Precisely by making use of some renormalization conditions, we calculate the one-loop path integral of canonical momentum. Moreover, we show that by introducing some new renormalization conditions, one can redefine the new degrees of freedom corresponding to a massive vector field.

scholarly journals Scalar–spinor fields interaction in de Sitter ambient space formalism

2019 ◽  
Vol 34 (25) ◽  
pp. 1950205 ◽  
Author(s):  
Y. Ahmadi ◽  
F. Jalilifard ◽  
M. V. Takook
Keyword(s):  
Scalar Field ◽  
Vector Fields ◽  
Ambient Space ◽  
Tree Level ◽  
Matrix Elements ◽  
De Sitter ◽  
Field Interaction ◽  
Spinor Fields ◽  
Space Formalism ◽  
Vector Formula

In de Sitter ambient space formalism, the massless minimally coupled scalar field can be constructed from a massless conformally coupled scalar field and a constant five-vector [Formula: see text]. Also, a constant five-vector [Formula: see text] appears in the interaction Lagrangian of massless minimally coupled scalar and spinor fields in this formalism. These constant five-vector fields can be fixed in the interaction case in the null curvature limit. Here, we will calculate the [Formula: see text] matrix elements of scalar–spinor field interaction in the tree level approximation. Then the constant five-vectors [Formula: see text] and [Formula: see text], will be fixed by comparing the [Formula: see text] matrix elements in the null curvature limits with the Minkowskian counterparts.

scholarly journals A path integral formulation for particle detectors: the Unruh-DeWitt model as a line defect

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Ivan M. Burbano ◽  
T. Rick Perche ◽  
Bruno de S. L. Torres
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Wilson Line ◽  
Transition Probability ◽  
Relativistic Quantum ◽  
Quantum Fields ◽  
Line Defect ◽  
Particle Detectors ◽  
Nonlocal Operators ◽  
Detector Model

Abstract Particle detectors are an ubiquitous tool for probing quantum fields in the context of relativistic quantum information (RQI). We formulate the Unruh-DeWitt (UDW) particle detector model in terms of the path integral formalism. The formulation is able to recover the results of the model in general globally hyperbolic spacetimes and for arbitrary detector trajectories. Integrating out the detector’s degrees of freedom yields a line defect that allows one to express the transition probability in terms of Feynman diagrams. Inspired by the light-matter interaction, we propose a gauge invariant detector model whose associated line defect is related to the derivative of a Wilson line. This is another instance where nonlocal operators in gauge theories can be interpreted as physical probes for quantum fields.

Path integral for the damped harmonic oscillator coupled to its dual

1994 ◽  
Vol 35 (3) ◽  
pp. 1185-1191 ◽  
Author(s):  
L. Chetouani ◽  
L. Guechi ◽  
T. F. Hammann ◽  
M. Letlout
Keyword(s):  
Harmonic Oscillator ◽  
Path Integral ◽  
Damped Harmonic Oscillator

scholarly journals EXTENDED FEYNMAN FORMULA FOR THE HARMONIC OSCILLATOR BY THE DISCRETE TIME METHOD

2010 ◽  
Vol 25 (03) ◽  
pp. 179-188
Author(s):  
KUNIO FUNAHASHI
Keyword(s):  
Harmonic Oscillator ◽  
Discrete Time ◽  
Path Integral ◽  
Feynman Formula ◽  
Discrete Formulation ◽  
Rigorous Method

We revisit the extended Feynman formula for the harmonic oscillator beyond and at caustics. The extension has been made by some authors, however, it is not obtained by the discrete formulation of path integral, which we consider the most reliable regularization of it. We derive the result by, especially at caustics, more rigorous method than previous.

scholarly journals SUPERSYMMETRIC CANONICAL COMMUTATION RELATIONS

2006 ◽  
Vol 21 (13n14) ◽  
pp. 2937-2951 ◽  
Author(s):  
FLORIN CONSTANTINESCU
Keyword(s):  
Vector Fields ◽  
Canonical Quantization ◽  
Canonical Commutation Relations ◽  
Commutation Relations

We discuss the unitarily-represented supersymmetric canonical commutation relations which are subsequently used to canonically quantize massive and massless chiral, antichiral and vector fields. The canonical quantization shows some new facets which do not appear in the nonsupersymmetric case. Our tool is the supersymmetric positivity generating the Hilbert–Krein structure of the N = 1 superspace.

Classical fields in curved spacetime

2021 ◽  
pp. 294-313
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Vector Fields ◽  
Lorentz Invariance ◽  
Conformal Symmetry ◽  
Scalar Fields ◽  
Gauge Fields ◽  
Curved Spacetime ◽  
Curved Space ◽  
Spinor Fields ◽  
Vector Gauge ◽  
Classical Fields

This chapter discusses classical fields in an arbitrary Riemann spacetime. General considerations are followed by the formulation of scalar fields with non-minimal coupling. Spontaneous symmetry breaking in curved space is shown to provide the induced gravity action with a cosmological constant. The construction of spinor fields in curved spacetime is based on the notions of group theory from Part I and on the local Lorentz invariance. Massless vector fields (massless vector gauge fields) are described and the interactions between scalar, fermion and gauge fields formulated. A detailed discussion of classical conformal transformations and conformal symmetry for both matter fields and vacuum action is also provided.

Field models

2021 ◽  
pp. 42-69
Author(s):  
Iosif L. Buchbinder ◽  
Ilya L. Shapiro
Keyword(s):  
Electromagnetic Field ◽  
Scalar Field ◽  
Spinor Field ◽  
Vector Fields ◽  
Sigma Model ◽  
Complex Scalar ◽  
Yang Mills ◽  
Scalar Field Models ◽  
Complex Scalar Field ◽  
Mills Field

This chapter provides constructions of Lagrangians for various field models and discusses the basic properties of these models. Concrete examples of field models are constructed, including real and complex scalar field models, the sigma model, spinor field models and models of massless and massive free vector fields. In addition, the chapter discusses various interactions between fields, including the interactions of scalars and spinors with the electromagnetic field. A detailed discussion of the Yang-Mills field is given as well.

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