Action Euclidean

Scalar Fields Consider a continuous field theory in Euclidean space with action... 

In this equation, Dx(x) and S [x(x)] are, respectively, the position space path measure and the Euclidean time action. The centroid density also formally defines a classical-like effective potential, i.e., ... 

A nice feature of (Euclidean) two-dimensional Yang-Mills gauge theory defined by the action... 

Instantons [18] correspond to the minima of the Euclidean action and are pseudoparticle solutions [19] of SU(2) Yang-Mills equations in Euclidean 4-space [20]. A complete construction for any Yang-Mills group is also available [21]. In other words [22, p. 80]... 

It is reasonable... to ask for the determination of the classical field configurations in Euclidean space which minimize the action, subject to appropriate asymptotic conditions in 4-space. These classical solutions are the instantons of the Yang-Mills theory. 

Again we use the Im F method in which the tunneling rate is determined by the nontrivial instanton paths that extremize the Euclidean action in the barrier. Suppose, for example, we let the potential V(Q) have a single minimum at Q = 0, V (O) = 0, separated from the continuous spectrum by a high barrier. The extrema of the action... 

In the light of the path integral representation for p(Q-,Q, [3), the latter may be semiclassically represented as proportional to exp[—Sj(Q )]. In this expression, Sl(Q ) is the Euclidean action on the /3-periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = () for T = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[-S2( ) )], where S2 is the action in the barrier for the closed straightforward trajectory which goes along the line with constant Q. ... 

Xe is the Euclidean action written for a classical trajectory corresponding to the sub-barrier rotation of the particle magnetization M. This is a typical instanton trajectory that satisfies the equations of motion... 

In such a way we have reduced our consideration to the problem of one-domain ferromagnetic particle with anisotropy energy described by equations (457) or (460). The problem of the macroscopic and mesoscopic quantum coherence which correspond to the tunnel switching of the magnetic moment between the two equilibrium directions was studied by Garg and Kim [332] in detail. This means that we may apply their results to calculate the tunneling frequency in the case of the hydrogen-bonded chain. Thus the Euclidean action SPe becomes... 

Figure 30. Role of the proton-phonon interaction in the coherent tunnel repolarization of a short chain. is the Euclidean renormalized action of phonons, and is their nonrenormalized action. The bond vibrations can effectively decrease the local proton tunneling (flo/tt < 0) and at the same time increase the coherent tunneling of protons ( 9 Et < // ).

Secondly, the commutator is the Lie product33 of the operators X Xs and Xu this choice of multiplication is particularly appropriate when one realizes that the X XS are the generators of the semisimple compact Lie group U , which is associated with the infinitesimal unitary transformations of the Euclidean vector space R (e.g., the space of the creation operators).34 With the preceding comments, the action of the transformation operator on the creation operators can formally be written in the usual form of the transformation law for covariant vectors,33... 

B. Fiedler and D. Turaev. Normal forms, resonances, and meandering tip motions near relative equilibria of Euclidean group actions. Arch. Ration. Mech. Anal., 145(2) 129-159, 1998. 

In general, it can be specified that the Euclidean time action functional for a reference system has a quadratic form in the path fluctuations variable q(r) such that... 

Geometry alone could not produce a theory of gravity, free of action at a distance, until physics managed to catch up with the ideas of Riemann. The development of special relativity, after discovery of the electromagnetic held, is described. It requires a holistic four-dimensional space-time, rather than three-dimensional Euclidean space and universal time. Accelerated motion, and therefore gravity, additionally requires this space-time to be non-Euclidean. The important conclusion is that relativity, more than a theory, is the only consistent description of physical reality at this time. Schemes for the unihed description of the gravitational and electromagnetic helds are briehy discussed. 

In order to define dynamical averages and to describe methods to calculate such quantities, we introduce a straightforward approach based on replacing dynamical averages with stationary averages of an observable that incorporates the action of the flow map. Let a and b be mappings from the phase space to the same Euclidean space, and define the temporal correlation of a and b (on the surface of energy E) by... 

A unitary operator leaves the lengths of vectors and angles between them unchanged. The transformations mediated through the action of such an operator are therefore analogous to those of rotations in three-dimensional Euclidean space. 

Here, the qs represents the subsystem coordinates, whereas the xs denotes the environment coordinates. 5 [ (.),x(.)] is the Euclidean action for the entire system (as described in Equation 11.11), which can be thought of as the summation of Euclidean actions for the subsystem, bath, and the interaction. The reduced density matrix is obtained from Equation 11.22 by integrating out the environmental coordinates, yielding... 

We now map this quantum phase transition onto a classical transition. Chakravarty and co-workers ° showed that the low-energy behavior of two-dimensional quantum Heisenberg antiferromagnets is generally described by a (2 - - l)-dimensional quantum rotor model with the Euclidean action... 

The Gamov exponent B is given by the Euclidean action, as first proposed by Gilbert ... 

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