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New systematic approximants are proposed for exponential functions, operators and inner derivation δ H . Remainders of systematic approximants are evaluated explicitly, which give degrees of convergence of approximants. The first approximant corresponds to Trotter's formula [1]: exp(^4 + J3) = lim [Qxp(A/n) Qxp(B/n)~] n . Some applications to physics are also discussed.

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A general scheme of fractal decomposition of exponential operators is presented in any order m. Namely, exp[x(A + B)] = S,n (x) + 0(x'" + i ) for any positive integer m, where s,, (x) = ,#A (peQ $5.. .$rrA with finite M depending on m. A general recursive scheme of construction of {t,) is given explicitly. It is proven that some of {t,} should be negative for m>3 and for any finite M (nonexistence theorem of positive decomposition), General systematic decomposition criterions based on a new type of time-ordering are also formulated. The decomposition exp [x(A + B) ] = [S,, (x/n) ] ' + 0(x" + l/n") yields a new efficient approach to quantum Monte Carlo simulations.

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The partition function of a quantal spin system is expressed by that of the Ising model, on the basis of the generalized Trotter formula. Thereby the ground state of the d-dimensional Ising model with a transverse field is proven to be equivalent to the (d+ 1)-dimensional Ising model at finite temperatures. A general relationship is established between the two partition functions of a general quantal spin system and the corresponding Ising model with many-spin interactions, which yields some rigorous results on quantum systems. Some applications are given. § 1. Introduction Critical phenomena in classical systems (whose Hamiltonians are described by scalar variables such as Ising spins) have been studied by many people since Onsager found the exact solution of the two-dimensional Ising model. It seems, however, very difficult to investigate critical phenomena in quantum systems such as the Heisenberg model. Although there are many perturbational calculations based on high temperature expansions, 1 } no rigorously soluble model of quantum systems •which show a phase transition at finite temperatures has been found except for extremely long-range interaction models!},s) Recently, Elliott, Pfeuty and W ood 4 } found numerically the equivalence of the ground state singularities of the d-dimensional Ising model with a transverse field 5 }, 6 } (which is the simplest quantal model) to the singularities of the (d + 1)dimensional Ising model. Quite recently Yanase, Takeshige and Suzuki 7 } have also confirmed numerically the above equivalence conjecture by calculating a power series expansion of the susceptibility up to a few more terms. On the other hand, the present author 8 } proved rigorously the equivalence of the two-dimensional Ising model to the ground state of the linear XY-model, which is described by the following Hamiltonians:

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