Periodic Path Integrals

2AA SURVEY ON MATSU BARA FREQUENCIES AND THE QUANTUM PERIODIC PATHS 

Since the actual path picture uses the periodic paths, they will be seen as the Fourier series (Feynman Hibbs, 19665 Feynman, 1972 Schulman, 1981 Wiegel, 1986 Kleinert, 2004 Putz, 2009) 

Moreover, under the condition the quantum paths (2.12) are real. 

With this, the quantified form of periodic path frequencies, Eq. (2.15), allows rewriting of the paths (2.12) under a separated form into constant and complex conjugated oscillating contributions 

Quantum Nanochemistty— Volume II Quantum Atoms and Periodicity 

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We have now all prerequisites to compute the Matsubara harmonic partition function (2.28) aiming to find out the Matsubara normalization of periodic path integrals. This will be addressed in the sequel. 

Being equipped with the periodic path integral teehnique we can present one of the most effieient ways for approximate the elfective-elassieal partition function (2.21) it firstly unfolds for a general external potential like the exact integral (Putz, 2009) ... 

Yet, the Feynman-Kleinert partition function is to be unfolded within the actual periodic path integral representation, with the help of Eqs. (2.28) and (2.50)... 

As a result of several complementary theoretical efforts, primarily the path integral centroid perspective [33, 34 and 35], the periodic orbit [36] or instanton [37] approach and the above crossover quantum activated rate theory [38], one possible candidate for a unifying perspective on QTST has emerged [39] from the ideas from [39, 40, 4T and 42]. In this theory, the QTST expression for the forward rate constant is expressed as [39]... 

The original idea of approximating the quantum mechanical partition function by a classical one belongs to Feynman [Feynman and Vernon 1963 Feynman and Kleinert 1986]. Expanding an arbitrary /S-periodic orbit, entering into the partition-function path integral, in a Fourier series in Matsubara frequencies v . 

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

About 50 years after Einstein, Gutzwiller applied the path integral method with a semiclassical approximation and succeeded to derive an approximate quantization condition for the system that has fully chaotic classical counterpart. His formula expresses the density of states in terms of unstable periodic orbits. It is now called the Gutzwiller trace formula [9,10]. In the last two decades, several physicists tested the Gutzwiiler trace formula for various... 

By comparing analogous terms in ( , x) and Q, we see that we can think of the partition function as a path integral over periodic orbits that recur in a complex time interval equal to i s flh/i = — ifih. There is no claim here that the closed paths used to generate Q correspond to actual quantum dynamics, but simply that there is an isomorphism. We therefore can refer to the equation above as the discretized path-integral (DPI) representation of the partition function. Using Feynman s notation, we have in the infinite-P limit... 

The DPI representation of the path integral that was developed in the preceding section is not unique. Another path-integral representation is often used that has come to be known as the Fourier representation [33,34,36-42,44,85]. Like the DPI representation, the Fourier representation transforms the path integral into an infinite-dimensional Riemann integral. In this formalism, we consider the paths to be periodic signals that can be represented as a Fourier series. Consider the density matrix p(x, x j8). Since the partition function is the trace of the density matrix, we have... 

To demonstrate how the transition probability density jy(x) can be used to compute Pj(0—the probability of being at milestone s at time t—it is convenient to define another fnnction Q t). It is the probability density that a trajectory will make a transition into s at time t. The normalization of Q,(t) is a little trickier compared to and is done with respect to trajectories. The set of events are trajectories of time periods between t and t + dt. Each of these trajectories pass (as a function of time) through milestones and is said to be in s if the last milestone it passes is s. The time spent at s (incubation time x) is set to zero when the trajectory passes s and the milestone it passes earlier was not s, (say s (s A s)). A fraction of the trajectories of time t + dt transition between s and s at the time interval [t, t + d/]. This fraction is denoted by QXOdt. It is explicitly given by a ratio of path integrals... 

The symbol = x(z ) in Eq. (2.21) denotes the fact that the path integral is performed over all paths that fulfill the periodicity x(0) = x(hp) of Eq. (2.13). Is now clear that we prefer to deal with such integrals because their completeness respecting with all possible (statistically closed) paths between two quantum events. Therefore, just for this path parameterization perspective the present level seems involving quite complex quantum phenomenology this will be further eiuiched in the sections to follow. 

Then, the associated atomie electronegativity and chemical hardness scales will be computed under general path integral quantum statistic framework and their periodic characteristics discussed respecting the general guidelines of the acceptability criteria and the finite-difierence counterparts. 

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