Quentin Smith
- Published in print:
- 1995
- Published Online:
- October 2011
- ISBN:
- 9780198263838
- eISBN:
- 9780191682650
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198263838.003.0012
- Subject:
- Religion, Philosophy of Religion, Theology
This chapter discusses the wave function of a Godless universe. It analyses the concept of nothingness, the interpretation of quantum cosmology, and Stephen Hawking's quasi-instrumentalist ...
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This chapter discusses the wave function of a Godless universe. It analyses the concept of nothingness, the interpretation of quantum cosmology, and Stephen Hawking's quasi-instrumentalist interpretation of quantum mechanics. It comments on the concept of imaginary time in Hawking's quantum cosmology and his physical interpretations of the exponential part of the wave function, and suggests that Hawking's quantum cosmology is rationally preferable to theism. It could be argued that his cosmology has greater predictive power since it predicts that it is highly probable there is a universe with properties we observe and that the universe is near the critical density.Less
This chapter discusses the wave function of a Godless universe. It analyses the concept of nothingness, the interpretation of quantum cosmology, and Stephen Hawking's quasi-instrumentalist interpretation of quantum mechanics. It comments on the concept of imaginary time in Hawking's quantum cosmology and his physical interpretations of the exponential part of the wave function, and suggests that Hawking's quantum cosmology is rationally preferable to theism. It could be argued that his cosmology has greater predictive power since it predicts that it is highly probable there is a universe with properties we observe and that the universe is near the critical density.
George Jaroszkiewicz
- Published in print:
- 2016
- Published Online:
- January 2016
- ISBN:
- 9780198718062
- eISBN:
- 9780191787553
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198718062.003.0021
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter discusses the role of imaginary time in physics. It emphasizes the difference between imaginary (or complex path) time and complex time. Starting with Minkowski’s famous 1908 lecture, ...
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This chapter discusses the role of imaginary time in physics. It emphasizes the difference between imaginary (or complex path) time and complex time. Starting with Minkowski’s famous 1908 lecture, the chapter considers the effect of replacing real time with imaginary time in relativity, Schrodinger wave mechanics, particle propagators, and Green’s functions in quantum field theory. The chapter reviews the use of imaginary time in Feynman path integrals, quantum gravity, quantum thermodynamics, and black hole thermodynamics.Less
This chapter discusses the role of imaginary time in physics. It emphasizes the difference between imaginary (or complex path) time and complex time. Starting with Minkowski’s famous 1908 lecture, the chapter considers the effect of replacing real time with imaginary time in relativity, Schrodinger wave mechanics, particle propagators, and Green’s functions in quantum field theory. The chapter reviews the use of imaginary time in Feynman path integrals, quantum gravity, quantum thermodynamics, and black hole thermodynamics.
Norman J. Morgenstern Horing
- Published in print:
- 2017
- Published Online:
- January 2018
- ISBN:
- 9780198791942
- eISBN:
- 9780191834165
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198791942.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the ...
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Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.Less
Multiparticle thermodynamic Green’s functions, defined in terms of grand canonical ensemble averages of time-ordered products of creation and annihilation operators, are interpreted as tracing the amplitude for time-developing correlated interacting particle motions taking place in the background of a thermal ensemble. Under equilibrium conditions, time-translational invariance permits the one-particle thermal Green’s function to be represented in terms of a single frequency, leading to a Lehmann spectral representation whose frequency poles describe the energy spectrum. This Green’s function has finite values for both t>t′ and t<t′ (unlike retarded Green’s functions), and the two parts G1> and G1< (respectively) obey a simple proportionality relation that facilitates the introduction of a spectral weight function: It is also interpreted in terms of a periodicity/antiperiodicity property of a modified Green’s function in imaginary time capable of a Fourier series representation with imaginary (Matsubara) frequencies. The analytic continuation from imaginary time to real time is discussed, as are related commutator/anticommutator functions, also retarded/advanced Green’s functions, and the spectral weight sum rule is derived. Statistical thermodynamic information is shown to be embedded in physical features of the one- and two-particle thermodynamic Green’s functions.
Tom Lancaster and Stephen J. Blundell
- Published in print:
- 2014
- Published Online:
- June 2014
- ISBN:
- 9780199699322
- eISBN:
- 9780191779435
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199699322.003.0026
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
There is a rather subtle connection between quantum field theory and statistical physics, and this is explored here, where the concepts of imaginary time and the Wick rotation are introduced.
There is a rather subtle connection between quantum field theory and statistical physics, and this is explored here, where the concepts of imaginary time and the Wick rotation are introduced.
George Jaroszkiewicz
- Published in print:
- 2016
- Published Online:
- January 2016
- ISBN:
- 9780198718062
- eISBN:
- 9780191787553
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198718062.003.0007
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter discusses the mathematics underpinning various models of time, starting with a review of the requirements that any reasonable model of time should satisfy. This is followed by a detailed ...
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This chapter discusses the mathematics underpinning various models of time, starting with a review of the requirements that any reasonable model of time should satisfy. This is followed by a detailed review of the continuum concept, because the most successful model of time we have is based on the real number continuum. Starting with the fundamental concept of a set, the chapter discusses ordered sets, functions, and linear continua. The chapter discusses the role of functions in predicting the future and retrodicting the past. Important mathematical concepts such as infinitesimals, signature, imaginary time, discrete time, fuzzy time, causal sets, and p-adic numbers are reviewed, as all of these have been used in one way or another to model time. The chapter comments on the difference between the mathematics that can be used on each side of the Heisenberg cut concept discussed in Chapter 2.Less
This chapter discusses the mathematics underpinning various models of time, starting with a review of the requirements that any reasonable model of time should satisfy. This is followed by a detailed review of the continuum concept, because the most successful model of time we have is based on the real number continuum. Starting with the fundamental concept of a set, the chapter discusses ordered sets, functions, and linear continua. The chapter discusses the role of functions in predicting the future and retrodicting the past. Important mathematical concepts such as infinitesimals, signature, imaginary time, discrete time, fuzzy time, causal sets, and p-adic numbers are reviewed, as all of these have been used in one way or another to model time. The chapter comments on the difference between the mathematics that can be used on each side of the Heisenberg cut concept discussed in Chapter 2.
