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Path integral in a magnetic field using the Trotter product formula

Bernard Gaveau, Mihoková, E, Roncadelli, M +1 more

The derivation of the Feynman path integral based on the Trotter product formula is extended to the case where the system is in a magnetic field.

Jan 1, 20048 citations

A dominated-type convergence theorem for the Feynman integral

G. W. Johnson, Jeong-Gyoo Kim

A dominated-type convergence theorem is proved for the operator-valued Feynman integral defined via the Trotter product formula. This definition is intimately related to the one of Feynman and the physical setting is the same as in his original paper. The convergence result given here provides the final piece of a unifying and rather satisfactory picture of three approaches to the Feynman integral.

Journal of Mathematical PhysicsMay 1, 20001 citation

Mathematics of the Feynman path integral

Vassili Nikitich Kolokoltsov

(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex

Jan 1, 20012 citations

The Jensen-Feynman inequality for path integrals extended to particles in a magnetic field: Application to a polaron in a magnetic field

J.T. Devresse, F. Brosens

Jan 1, 19940 citations

Remarks on the Time Transformation of Path-Integral Measures

Reijiro Kubo

By means of a time transformation of path-integral measures we rewrite the usual Feynman path integral as a path integral of determinants with the Wiener measure. We derive a formula, which is useful in evaluating path integrals.

Progress of Theoretical PhysicsFeb 1, 19870 citations

Path Integrals and Product Integrals

John F. Hamilton, Lawrence S. Schulman

The product integral is shown to be the appropriate mathematical tool for implementing the path integral for spin in its simplest form.

Journal of Mathematical PhysicsJan 1, 197118 citations

Path integral in a magnetic field using the Trotter product formula

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Path integral in a magnetic field using the Trotter product formula

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