Symplectic Geometry and Quantum Mechanics

Gosson, Maurice de

doi:10.1007/3-7643-7575-2

Cited by 346 publications

(625 citation statements)

References 105 publications

Supporting

Mentioning

620

Contrasting

Unclassified

Order By: Relevance

“…That we actually have equality is immediate, observing that we can translate the ball B(R) inside any cylinder with same radius, and that phase space translations are canonical transformations in their own right. More generally one calls symplectic capacity any function associating to subsets of phase space a non-negative number c( ), or +∞, and for which the properties listed in (12), (13), (14), and (15) are verified (see Hofer and Zehnder [14], Polterovich [28], or Schlenk [36] for the general theory of symplectic capacities; in [3,5] I have given a souped-down review of the topic). There exist infinitely many symplectic capacities, and the Gromov capacity is the smallest of all: c min ( ) ≤ c( ) for all and c. Is there a "biggest" symplectic capacity c max ?…”

Section: The Notion Of Symplectic Capacitymentioning

confidence: 99%

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Gosson

2009

Found Phys

112

View full text Add to dashboard Cite

We show that the strong form of Heisenberg's inequalities due to Robertson and Schrödinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the "minimum volume ellipsoid" together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what "quantum scale" is. We take the opportunity to discuss the principle of the symplectic camel, which is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics.

show abstract

Section: The Notion Of Symplectic Capacitymentioning

confidence: 99%

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Gosson

2009

Found Phys

112

View full text Add to dashboard Cite

show abstract

“…(20). The factorization of G as a product PS is closely related to the Iwasawa decomposition for a semisimple Lie group [30], which has played an important role in phase-space optics [31] and phase-space quantum mechanics [32]. However, the unique feature of the theory developed here is that the factorization is provided from the viewpoint of dynamics and selfconsistently constructed from the generalized envelope equation.…”

mentioning

confidence: 99%

Generalized Courant-Snyder Theory for Charged-Particle Dynamics in General Focusing Lattices

et al. 2013

View full text Add to dashboard Cite

The Courant-Snyder (CS) theory for one degree of freedom is generalized to the case of coupled transverse dynamics in general linear focusing lattices with quadrupole, skew-quadrupole, dipole, and solenoidal components, as well as torsion of the fiducial orbit and variation of beam energy. The envelope function is generalized into an envelope matrix, and the phase advance is generalized into a 4D sympletic rotation. The envelope equation, the transfer matrix, and the CS invariant of the original CS theory all have their counterparts, with remarkably similar expressions, in the generalized theory.

show abstract

“…is the Heisenberg operator; recall [14,26] that the action of T (z 0 ) on a function or distribution ψ is explicitly given by…”

Section: Born-jordan Pseudodifferential Operatorsmentioning

confidence: 99%

Born-Jordan pseudodifferential operators with symbols in the Shubin classes

Cordero

Gosson

Nicola

2017

Trans. Amer. Math. Soc. Ser. B

Self Cite

View full text Add to dashboard Cite

Abstract. We apply Shubin's theory of global symbol classes Γ m ρ to the Born-Jordan pseudodifferential calculus we have previously developed. This approach has many conceptual advantages and makes the relationship between the conflicting Born-Jordan and Weyl quantization methods much more limpid. We give, in particular, precise asymptotic expansions of symbols allowing us to pass from Born-Jordan quantization to Weyl quantization and vice versa. In addition we state and prove some regularity and global hypoellipticity results.

show abstract

Symplectic Geometry and Quantum Mechanics

Cited by 346 publications

References 105 publications

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

The Symplectic Camel and the Uncertainty Principle: The Tip of an Iceberg?

Generalized Courant-Snyder Theory for Charged-Particle Dynamics in General Focusing Lattices

Born-Jordan pseudodifferential operators with symbols in the Shubin classes

Contact Info

Product

Resources

About