Quantization of gauge field theories on the front form without gauge constraints: The Abelian case

Ovid C. Jacob

doi:10.1103/PhysRevD.50.5289

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Quantization of gauge field theories on the front form without gauge constraints: The Abelian case

Ovid C. Jacob

Stanford Linear Accelerator Center, Stanford University, Stanford, California 94309

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Phys. Rev. D 50, 5289 – Published 15 October, 1994

DOI: https://doi.org/10.1103/PhysRevD.50.5289

Abstract

Recently, we proposed a new front-form quantization which treated both the $x^{+}$ and the $x^{-}$ coordinates as front-form ‘‘times.’’ This quantization was found to preserve parity explicitly. In this paper we extend this construction to local Abelian gauge fields. We quantize this theory using a method proposed originally by Faddeev and Jackiw. We emphasize here the feature that quantizing along both $x^{+}$ and $x^{-}$ gauge theories does not require extra constraints (also known as ‘‘gauge conditions’’) to determine the solution uniquely.

References (21)

Kurt Sundermeyer, Constrained Dynamics (Springer Verlag, New York, 1982).
D. M. Gitman and I. V. Tyutin, Quantization of Fields with Constraints (Springer Verlag, New York, 1990).
Jan Govaerts, Hamiltonian Quantization and Constrained Dynamics (Leuven University, Leuven, Belgium, 1991).
Marc Henneaux and Claudio Teitelboim, Quantization of Gauge Systems (Princeton University, Princeton, NJ, 1992).
Ovid C. Jacob, ``Parity Conserving Light Cone Quantization,'' Report No. SLAC PUB 6188, hep th/9305076, 1993 (unpublished).
Ovid C. Jacob, ``Parity and Front Form Quantization of Field Theories,'' Report No. SLAC PUB 6391, hep th/9311016, 1993 (unpublished).
See also Mikolaj Sawicki and Dharam V. Ahluwalia, ``Parity Transformation in the Front Form,'' Report No. LA UR 93 4317, hep th 9312092 (unpublished).
A couple of recent papers also address the issue of consistency of field equations on the light cone [T. Heinzl and E. Werner, Regensburg Report No. TPR 93 3 (unpublished)] or full null plane field theory [Norbert E. Ligterink and B. L. G. Bakker, Vrije Universiteit, Amsterdam report, 1993 (unpublished)]. In both of these papers, the authors stay in the usual approach of taking only one light cone (null plane) ``time.'' The Regensburg paper actually makes some rather strong claims (with scant support) regarding lack of need for a second lightlike hyperplane, but they do it in the context of a mixed initial boundary value problem, whereas here we consider an initial value problem.
See also treatments of the Liouville field theory by E. D'Hoker and R. Jackiw [Phys. Rev. D 26, 3517 (1982)] and P. Mansfield [Nucl. Phys. B22, 419 (1983)].
R. Gambini and A. Restuccia, Phys. Rev. D 17, 3150 (1978).
R. K. Sachs, J. Math. Phys. 3, 908 (1962).
R. Penrose, Gen. Relativ. Gravit. 12, 225 (1980); Roger Penrose and Wolfgang Rindler, Spinors and Space Time (Cambridge University, New York, 1984), Vol. I. Chap. 5.
H. Müller zum Hagen and H. J. Seifert, Gen. Relativ. Gravit. 8, 259 (1977).
H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Proc. R. Soc. London Ser. A 269, 21 (1962).
L. D. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988).
R. Jackiw, ``(Constrained) Quantization Without Tears,'' Report No. CTP 2215, hep th/9306075 (unpublished).
Wei Min Zhang and Avaroth Harindranath, Phys. Rev. D 48, 4868 (1993) Phys. Lett. B 314, 223 (1993).
Note that the Fourier decomposition goes through even though we have two ``times.'' The point is that on each constant surface, be it xsup +=0 or $x^{-}$ =0, there is only one front form time, so that there is a consistent definition of a Fourier decomposition. On each of these surfaces, space time looks like 2+1 Minkowski space time.
Another solution is $A_{P}$ sup +=0= $A_{P}$ sup -, $C_{π}$ , $C_{ρ}$ arbitrary on xsup +=0. Then, by parity, on $x^{-}$ =0, we get $A^{-}$ sub M=0= $A_{M}$ sup +, so this makes Asup += $A^{-}$ =0, just like a usual gauge constraint. We do not want to explore this solution here, but rather are looking for other solutions. Or, take $C_{ρ}$ =0= $C_{π}$ on xsup +=0. We get $A_{P}$ sup +=[ ${partial}_{i}$ partialsup + /( ${partial}^{i}$ $)^{2}$ ] $A_{P}$ sup i + {[( partialsup + $)^{2}$ -2( ${partial}^{i}$ $)^{2}$ ]/( ${partial}^{i}$ $)^{4}$ } eψsub +sup dagger ψsub + and $A_{P}$ sup -=[ partial ${}^{i}$ ${partial}^{-}$ /( partial ${}^{i}$ $)^{2}$ ] $A_{P}$ sup i + {[( ${partial}^{-}$ $)^{2}$ -2 ( ${partial}^{i}$ $)^{2}$ ]/( ${partial}^{i}$ $)^{4}$ } eψsup ${dagger}_{-}$ $ψ_{-}$ . Finally, we can take $A_{P}$ sup - =0= $C_{ρ}$ on xsup +=0; this gives a contradiction, 0=2e $ψ_{-}$ sup dagger $ψ_{-}$ . Also, going to 1+1 dimensions, we get a result similar to the text in the limit of ${partial}_{i}$ to 0. So the solution in text is most consistent.
James D. Bjorken and Sidney D.Drell, Relativistic Quantum Fields (McGraw Hill, San Francisco, 1965), Chap. 15.
It seems that there might also be some problems with this approach as well once we consider derivative couplings like barψ $γ^{μ}$ $γ^{5}$ ψ ${partial}_{μ}$ φ where the fermions ψ couple to the pion field φ via an axial coupling. There might be some problems in implementing the reduced phase space quantization procedure in this case [Wei Mi Zhang (private communication)].

[c1] Kurt Sundermeyer, Constrained Dynamics (Springer Verlag, New York, 1982).

[c2] D. M. Gitman and I. V. Tyutin, Quantization of Fields with Constraints (Springer Verlag, New York, 1990).

[c3] Jan Govaerts, Hamiltonian Quantization and Constrained Dynamics (Leuven University, Leuven, Belgium, 1991).

[c4] Marc Henneaux and Claudio Teitelboim, Quantization of Gauge Systems (Princeton University, Princeton, NJ, 1992).

[c5] Ovid C. Jacob, ``Parity Conserving Light Cone Quantization,'' Report No. SLAC PUB 6188, hep th/9305076, 1993 (unpublished).

[c6] Ovid C. Jacob, ``Parity and Front Form Quantization of Field Theories,'' Report No. SLAC PUB 6391, hep th/9311016, 1993 (unpublished).

[c7] See also Mikolaj Sawicki and Dharam V. Ahluwalia, ``Parity Transformation in the Front Form,'' Report No. LA UR 93 4317, hep th 9312092 (unpublished).

[c8] A couple of recent papers also address the issue of consistency of field equations on the light cone [T. Heinzl and E. Werner, Regensburg Report No. TPR 93 3 (unpublished)] or full null plane field theory [Norbert E. Ligterink and B. L. G. Bakker, Vrije Universiteit, Amsterdam report, 1993 (unpublished)]. In both of these papers, the authors stay in the usual approach of taking only one light cone (null plane) ``time.'' The Regensburg paper actually makes some rather strong claims (with scant support) regarding lack of need for a second lightlike hyperplane, but they do it in the context of a mixed initial boundary value problem, whereas here we consider an initial value problem.

[c9] See also treatments of the Liouville field theory by E. D'Hoker and R. Jackiw [Phys. Rev. D 26, 3517 (1982)] and P. Mansfield [Nucl. Phys. B22, 419 (1983)].

[c10] R. Gambini and A. Restuccia, Phys. Rev. D 17, 3150 (1978).

[c11] R. K. Sachs, J. Math. Phys. 3, 908 (1962).

[c12] R. Penrose, Gen. Relativ. Gravit. 12, 225 (1980); Roger Penrose and Wolfgang Rindler, Spinors and Space Time (Cambridge University, New York, 1984), Vol. I. Chap. 5.

[c13] H. Müller zum Hagen and H. J. Seifert, Gen. Relativ. Gravit. 8, 259 (1977).

[c14] H. Bondi, M. G. J. van der Burg, and A. W. K. Metzner, Proc. R. Soc. London Ser. A 269, 21 (1962).

[c15] L. D. Faddeev and R. Jackiw, Phys. Rev. Lett. 60, 1692 (1988).

[c16] R. Jackiw, ``(Constrained) Quantization Without Tears,'' Report No. CTP 2215, hep th/9306075 (unpublished).

[c17] Wei Min Zhang and Avaroth Harindranath, Phys. Rev. D 48, 4868 (1993) Phys. Lett. B 314, 223 (1993).

[c18] Note that the Fourier decomposition goes through even though we have two ``times.'' The point is that on each constant surface, be it xsup +=0 or $x^{-}$ =0, there is only one front form time, so that there is a consistent definition of a Fourier decomposition. On each of these surfaces, space time looks like 2+1 Minkowski space time.

[c19] Another solution is $A_{P}$ sup +=0= $A_{P}$ sup -, $C_{π}$ , $C_{ρ}$ arbitrary on xsup +=0. Then, by parity, on $x^{-}$ =0, we get $A^{-}$ sub M=0= $A_{M}$ sup +, so this makes Asup += $A^{-}$ =0, just like a usual gauge constraint. We do not want to explore this solution here, but rather are looking for other solutions. Or, take $C_{ρ}$ =0= $C_{π}$ on xsup +=0. We get $A_{P}$ sup +=[ ${partial}_{i}$ partialsup + /( ${partial}^{i}$ $)^{2}$ ] $A_{P}$ sup i + {[( partialsup + $)^{2}$ -2( ${partial}^{i}$ $)^{2}$ ]/( ${partial}^{i}$ $)^{4}$ } eψsub +sup dagger ψsub + and $A_{P}$ sup -=[ partial ${}^{i}$ ${partial}^{-}$ /( partial ${}^{i}$ $)^{2}$ ] $A_{P}$ sup i + {[( ${partial}^{-}$ $)^{2}$ -2 ( ${partial}^{i}$ $)^{2}$ ]/( ${partial}^{i}$ $)^{4}$ } eψsup ${dagger}_{-}$ $ψ_{-}$ . Finally, we can take $A_{P}$ sup - =0= $C_{ρ}$ on xsup +=0; this gives a contradiction, 0=2e $ψ_{-}$ sup dagger $ψ_{-}$ . Also, going to 1+1 dimensions, we get a result similar to the text in the limit of ${partial}_{i}$ to 0. So the solution in text is most consistent.

[c20] James D. Bjorken and Sidney D.Drell, Relativistic Quantum Fields (McGraw Hill, San Francisco, 1965), Chap. 15.

[c21] It seems that there might also be some problems with this approach as well once we consider derivative couplings like barψ $γ^{μ}$ $γ^{5}$ ψ ${partial}_{μ}$ φ where the fermions ψ couple to the pion field φ via an axial coupling. There might be some problems in implementing the reduced phase space quantization procedure in this case [Wei Mi Zhang (private communication)].

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Quantization of gauge field theories on the front form without gauge constraints: The Abelian case

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