Variational calculations and Bogolubov transformations

José Wudka

doi:10.1103/PhysRevD.39.3000

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Variational calculations and Bogolubov transformations

José Wudka

Physics Department, University of California at Davis, Davis, California 95616

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Phys. Rev. D 39, 3000 – Published 15 May, 1989

DOI: https://doi.org/10.1103/PhysRevD.39.3000

Abstract

The usefulness of the Bogolubov transformations in variational calculations is demonstrated using the massive Thirring model as a specific example. It is shown that such transformations can be chosen so that the expectation value of the Hamiltonian is regulator independent when the exact form of the counterterms is used. Moreover enough freedom is left to optimize the variational state as a trial vacuum. The problem of dealing with the anomalous commutators that arise in the model is discussed.

References (22)

N. N. Bogolubov, Zh. Eksp. Teor. Fiz. 34 58 (1958) [Sov. Phys. JETP 7, 51 (1958)].
W. E. Thirring, Ann. Phys. (N.Y.) 3, 91 (1958); C. M. Sommerfield, ibid. 26, 1 (1963); B. Klaiber, in Quantum Theory and Statistical Theory, 1967 Boulder Lectures in Physics, edited by W. E. Brittin et al. (Gordon and Breach, New York, 1968), Vol. XA.
K. Johnson, Nuovo Cimento 20, 773 (1961).
S. Coleman, Phys. Rev. D 11, 3424 (1975); S. Mandelstam, ibid. 11, 3026 (1975).
We shall refer to LAMBDA as an ultraviolet cutoff. When required, such a cutoff will be implemented using Pauli-Villars regulators of mass LAMBDA.
K. Wilson, Phys. Rev. D 2, 1473 (1970); A. H. Mueller and T. L. Trueman, ibid. 4, 1635 (1971); R. Jackiw, ibid. 3, 2005 (1971); M. Gomes and J. H. Lowenstein, Nucl. Phys. B45, 252 (1972); A. Roy Chowdhury et al., J. Phys. A 9, 1191 (1976); S. Aoyama, Nuovo Cimento 55A, 409 (1980).
T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); J. I. Latorre and J. Soto, ibid. 34, 3111 (1986); P. M. Stevenson and G. A. Hajj, ibid. 34, 3117 (1986).
This was also used by A. H. Mueller and T. L. Trueman, Phys. Rev. D 4, 1635 (1971).
J. Schwinger, Phys. Rev. 128, 2425 (1962).
For reviews, see R. Jackiw, in The Laws of Hadronic Matter, edited by A. Zichichi (Associated, New York, London, 1975); in Current Algebra and Anomalies (World Scientific, Singapore, 1985).
J. D. Bjorken, Phys. Rev. 148, 1467 (1966); K. Johnson and F. E. Low, Prog. Theor. Phys. (Kyoto), Suppl. 37, 74 (1966). For a review, see Jackiw, in Current Algebra and Anomalies (Ref. 10).
B. Schroer and T. Truong, Phys. Rev. D 15, 1684 (1977).
This is a standard method used in statistical physics for extracting the vacuum energy. See, for example, A. A. Abrikosov et al., Methods of Quantum Field Theory in Statistical Mechanics (Dover, New York, 1975), Sec. 10.3.
There is one minor problem: if we expand the asymptotic form of Q in powers of g, a k-dependent contribution appears to first order, while no such term appears in a perturbative evaluation. This need not be a problem since the annoying terms can be canceled by subleading contributions to Q. We believe this to be the case, though we have not rigorously proved so.
The poles in g are of no concern since the corresponding residues are LAMBDA independent; therefore, they are canceled by the finite (i.e., LAMBDA-independent) terms which we do not display.
S. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1959).
C. Itzykson and J-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
J. Wudka, Phys. Rev. D 37, 1464 (1988).
In the above manipulations we assumed at first δ > 3 to obtain convergent results; we then continued the result to all g < π /2. We will continue to use this procedure in calculation B and C, though we will not explicitly mention it.
See, for example, P. West, Introduction to Supersymmetry and Supergravity (World Scientific, Singapore, 1986), Chap. 18, and references therein.
R. P. Feynman, Nucl. Phys. B188, 479 (1981).
L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Pergamon, Oxford, 1978), Sec. 12.

[c1] N. N. Bogolubov, Zh. Eksp. Teor. Fiz. 34 58 (1958) [Sov. Phys. JETP 7, 51 (1958)].

[c2] W. E. Thirring, Ann. Phys. (N.Y.) 3, 91 (1958); C. M. Sommerfield, ibid. 26, 1 (1963); B. Klaiber, in Quantum Theory and Statistical Theory, 1967 Boulder Lectures in Physics, edited by W. E. Brittin et al. (Gordon and Breach, New York, 1968), Vol. XA.

[c3] K. Johnson, Nuovo Cimento 20, 773 (1961).

[c4] S. Coleman, Phys. Rev. D 11, 3424 (1975); S. Mandelstam, ibid. 11, 3026 (1975).

[c5] We shall refer to LAMBDA as an ultraviolet cutoff. When required, such a cutoff will be implemented using Pauli-Villars regulators of mass LAMBDA.

[c6] K. Wilson, Phys. Rev. D 2, 1473 (1970); A. H. Mueller and T. L. Trueman, ibid. 4, 1635 (1971); R. Jackiw, ibid. 3, 2005 (1971); M. Gomes and J. H. Lowenstein, Nucl. Phys. B45, 252 (1972); A. Roy Chowdhury et al., J. Phys. A 9, 1191 (1976); S. Aoyama, Nuovo Cimento 55A, 409 (1980).

[c7] T. Barnes and G. I. Ghandour, Phys. Rev. D 22, 924 (1980); J. I. Latorre and J. Soto, ibid. 34, 3111 (1986); P. M. Stevenson and G. A. Hajj, ibid. 34, 3117 (1986).

[c8] This was also used by A. H. Mueller and T. L. Trueman, Phys. Rev. D 4, 1635 (1971).

[c9] J. Schwinger, Phys. Rev. 128, 2425 (1962).

[c10] For reviews, see R. Jackiw, in The Laws of Hadronic Matter, edited by A. Zichichi (Associated, New York, London, 1975); in Current Algebra and Anomalies (World Scientific, Singapore, 1985).

[c11] J. D. Bjorken, Phys. Rev. 148, 1467 (1966); K. Johnson and F. E. Low, Prog. Theor. Phys. (Kyoto), Suppl. 37, 74 (1966). For a review, see Jackiw, in Current Algebra and Anomalies (Ref. 10).

[c12] B. Schroer and T. Truong, Phys. Rev. D 15, 1684 (1977).

[c13] This is a standard method used in statistical physics for extracting the vacuum energy. See, for example, A. A. Abrikosov et al., Methods of Quantum Field Theory in Statistical Mechanics (Dover, New York, 1975), Sec. 10.3.

[c14] There is one minor problem: if we expand the asymptotic form of Q in powers of g, a k-dependent contribution appears to first order, while no such term appears in a perturbative evaluation. This need not be a problem since the annoying terms can be canceled by subleading contributions to Q. We believe this to be the case, though we have not rigorously proved so.

[c15] The poles in g are of no concern since the corresponding residues are LAMBDA independent; therefore, they are canceled by the finite (i.e., LAMBDA-independent) terms which we do not display.

[c16] S. Adler and W. A. Bardeen, Phys. Rev. 182, 1517 (1959).

[c17] C. Itzykson and J-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).

[c18] J. Wudka, Phys. Rev. D 37, 1464 (1988).

[c19] In the above manipulations we assumed at first δ > 3 to obtain convergent results; we then continued the result to all g < π /2. We will continue to use this procedure in calculation B and C, though we will not explicitly mention it.

[c20] See, for example, P. West, Introduction to Supersymmetry and Supergravity (World Scientific, Singapore, 1986), Chap. 18, and references therein.

[c21] R. P. Feynman, Nucl. Phys. B188, 479 (1981).

[c22] L. D. Landau and E. M. Lifshitz, Mechanics, 3rd ed. (Pergamon, Oxford, 1978), Sec. 12.

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Variational calculations and Bogolubov transformations

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