Taro Kashiwa

Coherent states, Path integral, and Semiclassical approximation

Using the generalized coherent states it is shown that the path integral formulas for SU(2) and SU(1,1) (in the discrete series) are WKB exact, if it is started from the trace of e−iTĤ, where H is given by a linear combination of generators. In this case, the WKB approximation is achieved by taking a large ‘‘spin’’ limit: J,K→∞, under which it is found that each coefficient vanishes except the leading term which indeed gives the exact result. It is further pointed out that the discretized form of path integral is indispensable, in other words, the continuum path integral expression sometimes leads to a wrong result. Therefore great care must be taken when some geometrical action would be adopted, even if it is so beautiful as the starting ingredient of path integral. Discussions on generalized coherent states are also presented both from geometrical and simple oscillator (Schwinger boson) points of view.

Coherent states over Grassmann manifolds and the WKB exactness in path integral

U(N) coherent states over Grassmann manifold, GN,n≂U(N)/(U(n)×U(N−n)), are formulated to be able to argue the WKB exactness in the path integral representation of a character formula. The phenomena is the so‐called localization of Duistermaat–Heckman. The exponent in the path integral formula is proportional to an integer k labeling the U(N) representation. Thus, when k→∞ a usual semiclassical approximation, by regarding k∼1/ℏ, can be performed to yield a desired conclusion. The mechanism of the localization is uncovered by the help of the (generalized) Schwinger boson technique. The discussion on the Feynman kernel is also presented.

Exactness in the Wentzel–Kramers–Brillouin approximation for some homogeneous spaces

Analysis of the Wentzel–Kramers–Brillouin (WKB) exactness in some homogeneous spaces is attempted. CPN as well as its noncompact counterpart DN,1 is studied. U(N+1) or U(N,1) based on the Schwinger bosons leads us to CPN or DN,1 path integral expression for the quantity tr e−iHT, with the aid of coherent states. The WKB approximation terminates in the leading order and yields the exact result provided that the Hamiltonian is given by a bilinear form of the creation and the annihilation operators. An argument on the WKB exactness to more general cases is also made.

Path integration on spheres: Hamiltonian operators from the Faddeev-Senjanovic path integral formula

Abstract A study is made of the path integral quantization for a D -dimensional sphere as an example of constrained systems by means of the Faddeev-Senjanovic formula, although no one has checked its validity with paying attentions to a global structure of integration regions. By adopting a well-defined path measure obtained through the time discretization and applying formulas of spherical harmonics, operator Hamiltonians can be picked out from their formula without recourse to any approximation such as a semi-classical one ( h → 0). The analysis is made with a model, H=C p 2 +M 2 C 2 . We might thus regard the path integral formula of Faddeev-Senjanovic as a powerful breakthrough to “quantum” constrained systems where ordinary canonical approaches sometimes encounter difficulties in giving well-defined canonical momenta because of the uncertainty principle.

More about path integrals for spin

Abstract The path integral for the SU (2) spin system is reconsidered. We show that the Nielsen-Rohrlich (NR) formula is equivalent to the spin coherent state expression so that the phase space in the NR formalism is not topologically nontrivial. We also perform the WKB approximation in the NR formula and find that it gives the exact result.

Auxiliary field method as a powerful tool for nonperturbative study

The auxiliary field method, defined through introducing an auxiliary (also called as the Hubbard-Stratonovich or the Mean-) field and utilizing a loop-expansion, gives an excellent result for a wide range of a coupling constant. The analysis is made for Anharmonic-Oscillator and Double-Well examples in 0-(a simple integral) and 1-(quantum mechanics)dimension. It is shown that the result becomes more and more accurate by taking a higher loop into account in a weak coupling region, however, it is not the case in a strong coupling region. The 2-loop approximation is shown to be still insufficient for the Double-Well case in quantum mechanics.

Proof for Gauge Independence of the Energy‐Momentum Tensor in Quantum Electrodynamics

Proof is given for gauge independence of the (Belinfantes) symmetric energy-momentum tensor in QED. Under the covariant LSZ-formalism it is shown that expectation values, supplemented with physical state conditions, of the energy-momentum tensor are gauge independent to all orders of the purturbation theory (the loop expansion). A study is also made, in terms of the gauge invariant operators of electron (known as the Diracs or Steinmanns electron) and photon, in expectation of gauge invariant result without any restriction. It is, however, shown that singling out gauge invariant quantities is merely synonymous to fixing a gauge, then there needs again a use of the asymptotic condition to obtain gauge independent results.

Canonical Transformations in the Path Integral and the Operator Formalism

Abstract A general framework of canonical transformations in the path integral is presented under the guidance of classical mechanics. The master equation is given in terms of the difference instead of the differential in classical mechanics. In the case of infinitesimal transformations the operator counterpart is found to be a PQ -ordered form of the classical generator. A path integral formula for polar coordinates is also discussed to be understood as a faithful representation of the Schrodinger equation.

Gauge independence in terms of the functional integral

A gauge-invariant formulation in quantum electrodynamics, characterized by an arbitrary function {phi}{sub {mu}}(x), is reconsidered. Operators in a covariant case, however, are ill defined because of a {phi}{sub {mu}}(x){approximately}{partial_derivative}{sub {mu}}/{open_square}-type singularity in Minkowski space. We then build up a Euclidean path integral formula, starting with a noncovariant but well-defined canonical operator formalism. The final expression is covariant, free from the pathology, and shows that the model can be interpreted as the {phi}{sub {mu}}-gauge fixing. Utilizing this formula we prove the gauge independence of the free energy as well as the S matrix. We also clarify the reason why it is so simple and straightforward to perform gauge transformations in the path integral. {copyright} {ital 1997} {ital The American Physical Society}

Ordering, Symbols and Finite-Dimensional Approximations of Path Integrals

We derive general form of finite-dimensional approximations of path integrals for both bosonic and fermionic canonical systems in terms of symbols of operators determined by operator ordering. We argue that for a system with a given quantum Hamiltonian such approximations are independent of the type of symbols up to terms of