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Register a new userGeometrical Properties of Feynman Path Integrals
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Authors
Timur F. Kamalov
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This model is one of the possible geometrical interpretations of Quantum Mechanics where found to every image Path correspondence the geodesic trajectory of classical test particles in the random geometry of the stochastic fields background. We are finding to the imagined Feynman Path a classical model of test particles as geodesic trajectory in the curved space of Projected Hilbert space on Blochs sphere.
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A method for reducing Feynman integrals, depending on several kinematic variables and masses, to a combination of integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle and box integrals with massless internal propagators to simpler integrals is described in detail. The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the $_2F_1$ Gauss hypergeometric function and the logarithmic function was derived. By using the functional equations, the one-loop box integral with massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the $F_1$ Appell and $_2F_1$ Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension $d$. The reduction equations for the box integral with some kinematic variables equal to zero are considered.
The roles of Lie groups in Feynmans path integrals in non-relativistic quantum mechanics are discussed. Dynamical as well as geometrical symmetries are found useful for path integral quantization. Two examples having the symmetry of a non-compact Lie group are considered. The first is the free quantum motion of a particle on a space of constant negative curvature. The system has a group SO(d,1) associated with the geometrical structure, to which the technique of harmonic analysis on a homogeneous space is applied. As an example of a system having a non-compact dynamical symmetry, the d-dimensional harmonic oscillator is chosen, which has the non-compact dynamical group SU(1,1) besides its geometrical symmetry SO(d). The radial path integral is seen as a convolution of the matrix functions of a compact group element of SU(1,1) on the continuous basis.
Whenever variables $phi=(phi^1,phi^2,ldots)$ are discarded from a system, and the discarded information capacity $mathcal{S}(x)$ depends on the value of an observable $x$, a quantum correction $Delta V_mathrm{eff}(x)$ appears in the effective potential [arXiv:1707.05789]. Here I examine the origins and implications of $Delta V_mathrm{eff}$ within the path integral, which I construct using Synges world function. I show that the $phi$ variables can be `integrated out of the path integral, reducing the propagator to a sum of integrals over observable paths $x(t)$ alone. The phase of each path is equal to the semiclassical action (divided by $hbar$) including the same correction $Delta V_mathrm{eff}$ as previously derived. This generalises the prior results beyond the limits of the Schrodinger equation; in particular, it allows us to consider discarded variables with a history-dependent information capacity $mathcal{S}=mathcal{S}(x,int^t f(x(t))mathrm{d} t)$. History dependence does not alter the formula for $Delta V_mathrm{eff}$.
Although the path-integral formalism is known to be equivalent to conventional quantum mechanics, it is not generally obvious how to implement path-based calculations for multi-qubit entangled states. Whether one takes the formal view of entangled states as entities in a high-dimensional Hilbert space, or the intuitive view of these states as a connection between distant spatial configurations, it may not even be obvious that a path-based calculation can be achieved using only paths in ordinary space and time. Previous work has shown how to do this for certain special states; this paper extends those results to all pure two-qubit states, where each qubit can be measured in an arbitrary basis. Certain three-qubit states are also developed, and path integrals again reproduce the usual correlations. These results should allow for a substantial amount of conventional quantum analysis to be translated over into a path-integral perspective, simplifying certain calculations, and more generally informing research in quantum foundations.
We discuss the time-continuous path integration in the coherent states basis in a way that is free from inconsistencies. Employing this notion we reproduce known and exact results working directly in the continuum. Such a formalism can set the basis to develop perturbative and non-perturbative approximations already known in the quantum field theory community. These techniques can be proven useful in a great variety of problems where bosonic Hamiltonians are used.
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