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## OUTLOOK

*Francesco Calogero*

### in Isochronous Systems

- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199535286
- eISBN:
- 9780191715853
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199535286.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In Chapter 8 five directions of future research are tersely mentioned in order of increasing importance. The investigation of Diophantine relations, such as those reported in Appendix C. The ... More

## Tonelli Lagrangians and Hamiltonians on Compact Manifolds

*Alfonso Sorrentino*

### in Action-minimizing Methods in Hamiltonian Dynamics (MN-50): An Introduction to Aubry-Mather Theory

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164502
- eISBN:
- 9781400866618
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164502.003.0001
- Subject:
- Mathematics, Applied Mathematics

This chapter introduces the basic setting: Tonelli Lagrangians and Hamiltonians on a compact manifold. It discusses their main properties and some examples, and provides the opportunity to recall ... More

## BOSONIZATION

*Thierry Giamarchi*

### in Quantum Physics in One Dimension

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198525004
- eISBN:
- 9780191711909
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525004.003.0002
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter explains bosonization, a useful technique for describing the low-energy properties of one-dimensional systems. Bosonization is introduced by presenting a model on which this method is ... More

## PATH INTEGRALS IN PHASE SPACE

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter contains a few additional results such as a definition of path integrals over phase space trajectories, and the problems generated by the quantization of lagrangians with potentials ... More

## PATH INTEGRALS IN QUANTUM MECHANICS

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0002
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter constructs the path integral associated with the statistical operator e -βH in the case of Hamiltonians of the simple form p2/2m + V (q). The path ... More

## PARTITION FUNCTION AND SPECTRUM

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

A direct application of the calculation of the partition function is provided by determining the spectrum of a Hamiltonian (which is assumed to be discrete, for simplicity). In this chapter, only ... More

## PATH INTEGRALS AND QUANTIZATION

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0005
- Subject:

This chapter constructs path integrals for general Hamiltonians with potentials linear in the velocities, like Hamiltonians of particles in a magnetic field. Two examples are considered: a quantum ... More

## PATH INTEGRALS AND HOLOMORPHIC FORMALISM

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0006
- Subject:

This chapter introduces the holomorphic representation of quantum mechanics, because it allows a study of the properties of boson systems both from the point of view of evolution and of quantum ... More

## Weak KAM Theory and Forcing Equivalence

*Kaloshin Vadim and Zhang Ke*

### in Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom: (AMS-208)

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter describes weak Kolmogorov-Arnold-Moser (KAM) theory and forcing relation. One change from the standard presentation is that one needs to modify the definition of Tonelli Hamiltonians to ... More

## Aubry-Mather type at the single resonance

*Kaloshin Vadim and Zhang Ke*

### in Arnold Diffusion for Smooth Systems of Two and a Half Degrees of Freedom: (AMS-208)

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691202525
- eISBN:
- 9780691204932
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202525.003.0009
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter proves Aubry-Mather type in the single-resonance regime. It proves that for a single-resonance normal form system which satisfies the non-degeneracy conditions, every c in the resonance ... More

## Dynamics IV: Aspects of locality: clustering, microcausality, and analyticity

*Anthony Duncan*

### in The Conceptual Framework of Quantum Field Theory

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199573264
- eISBN:
- 9780191743313
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199573264.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Chapter 5 discussed two possible approaches to constructing a relativistically invariant theory of particle scattering. The first attempt — a frontal assault in which one directly writes down for ... More

## A brief introduction to strongly correlated electronic materials

*E. Dagotto and Y. Tokura*

### in Multifunctional Oxide Heterostructures

- Published in print:
- 2012
- Published Online:
- January 2013
- ISBN:
- 9780199584123
- eISBN:
- 9780191745331
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199584123.003.0001
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter summarizes the status of the very active field of research where the strength of the effective Coulombic repulsion among electrons is comparable to the bandwidth of the carriers. It ... More

## Absolute time

*George Jaroszkiewicz*

### in Images of Time: Mind, Science, Reality

- Published in print:
- 2016
- Published Online:
- January 2016
- ISBN:
- 9780198718062
- eISBN:
- 9780191787553
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198718062.003.0014
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses Newton’s concept of absolute time and its use in classical mechanics. It defines clocks and our freedom to change them, that is, reparametrization of time. The chapter goes on ... More

## The renormalization group (RG) approach: The critical theory near four dimensions

*Jean Zinn-Justin*

### in Quantum Field Theory and Critical Phenomena: Fifth Edition

- Published in print:
- 2021
- Published Online:
- June 2021
- ISBN:
- 9780198834625
- eISBN:
- 9780191872723
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198834625.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In Chapter 14, the singular behavior of ferromagnetic systems with O(N) symmetry and short-range interactions, near a second order phase transition has been determined in the mean-field ... More

## Quantum mechanics (QM): Path integrals in phase space

*Jean Zinn-Justin*

### in Quantum Field Theory and Critical Phenomena: Fifth Edition

- Published in print:
- 2021
- Published Online:
- June 2021
- ISBN:
- 9780198834625
- eISBN:
- 9780191872723
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198834625.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

In Chapter 2, a path integral representation of the quantum operator e-β H in the case of Hamiltonians H of the separable form p2/2m + V(q) has been constructed. Here, the construction is extended to ... More

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