Abstract
We show that the method of finite elements reduces intractable quantum operator differential equations to completely solvable operator difference equations. Early work suggests that this approximation technique is extremely accurate and very well suited to algebraic manipulation on a computer.
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This talk is based on published as well as unpublished work: C. M. Bender and D. H. Sharp, Phys. Rev. Lett. 50, 1535 (1983).
C. M. Bender, K. A. Milton, and D. H. Sharp, Phys. Rev. Lett. 51, 1815 (1983); C. M. Bender, K. A. Milton, and D. H. Sharp, “Gauge Invariance and the Finite-Element Solution of the Schwinger Model,” submitted to Phys. Rev.
Useful general references on the finite element method are G. Strang and G. J. Fix, An Analysis of the Finite Element Method, (Prentice-Hall, Inc., Englewood Cliffs, 1973).
T. J. Chung, Finite Element Analysis in Fluid Dynamics (McGraw-Hill, New York, 1978).
There are several interesting remarks to be made here. One intriguing question is whether (12) and (13) might be used in combination with [q0, p0] = i to find a spectrum generating algebra. Second, one may ask what happens when the equation y = g (x) has multiple roots; that is, what role is played by instantons In these lattice calculations?
The matrix S is a numerical matrix containing the lattice spacings h and k. It is symmetric because with properly chosen boundary conditions the operator ∇2 the continuum is symmetric.
For a detailed discussion see L. H. Karsten and J. Smit, Nucl. Phys. B183, 103 (1981).
H. B. Nielsen and M. Ninomiya, Nucl. Phys. B185, 20 (1981).
J. M. Rabin, Phys. Rev. D24, 3218 (1981).
By experimenting with various types of difference schemes, R. Stacey independently discovered the dispersion relation (42) and the Kogut-Susklnd version of (41). See Phys. Rev. D26, 468 (1982).
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© 1985 Kluwer Academic Publishers
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Bender, C.M. (1985). A Proposal for the Solution of Quantum Field Theory Problems Using a Finite-Element Approximation. In: Pavelle, R. (eds) Applications of Computer Algebra. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-6888-5_18
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DOI: https://doi.org/10.1007/978-1-4684-6888-5_18
Publisher Name: Springer, Boston, MA
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