A Proposal for the Solution of Quantum Field Theory Problems Using a Finite-Element Approximation

  • Carl M. Bender


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.


Dirac Equation dIscrete Action Heisenberg Equation Equal Time Commutation Relation Fermion Doubling 
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  1. 1.
    This talk is based on published as well as unpublished work: C. M. Bender and D. H. Sharp, Phys. Rev. Lett. 50, 1535 (1983).MathSciNetCrossRefGoogle Scholar
  2. 1a.
    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.MathSciNetCrossRefGoogle Scholar
  3. 2.
    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).MATHGoogle Scholar
  4. 2a.
    T. J. Chung, Finite Element Analysis in Fluid Dynamics (McGraw-Hill, New York, 1978).MATHGoogle Scholar
  5. 3.
    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?Google Scholar
  6. 4.
    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.Google Scholar
  7. 5.
    For a detailed discussion see L. H. Karsten and J. Smit, Nucl. Phys. B183, 103 (1981).CrossRefGoogle Scholar
  8. 5a.
    H. B. Nielsen and M. Ninomiya, Nucl. Phys. B185, 20 (1981).MathSciNetCrossRefGoogle Scholar
  9. 5b.
    J. M. Rabin, Phys. Rev. D24, 3218 (1981).Google Scholar
  10. 6.
    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).Google Scholar

Copyright information

© Kluwer Academic Publishers 1985

Authors and Affiliations

  • Carl M. Bender
    • 1
  1. 1.Department of PhysicsWashington UniversitySt. LouisUSA

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