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I have learnt field-antifield quantisation and know that it can be used for very general gauge theories - open and reducible. I have not got much into light-cone quantisation but I am unable to see the motivation of why we need light cone quantisation if field-antifield is so powerful. Can you point out difference between light-cone and field-antifield quantisation and also which is more powerful?

Qmechanic
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Bhavya Bhatt
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1 Answers1

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  1. If we are tasked with quantizing a classical gauge theory, we want to try the lightcone (LC) quantization first because it is so much simpler. In this way we can separate out & count physical DOFs, and otherwise familiarize ourselves with the quantum theory at hand.

  2. However, LC quantization sacrifices manifest Lorentz symmetry, so ultimately we would want to use covariant BRST quantization.

  3. The most general Lagrangian BRST quantization is the Batalin-Vilkovisky (BV) field-antifield formalism, which in principle can be applied to the most general open and reducible gauge algebra. It has been successfully used in a long list of gauge theories, see also e.g. this & this Phys.SE posts.

    However, the BV field-antifield formalism may break down for various reasons. E.g.

    • The original paper (Ref. 1) is using the deWitt condensed notation $$\varphi^{\alpha}(x)~\longrightarrow~\varphi^i,$$ where spacetime-indices are suppressed, heuristically treating infinite-dimensional field configurations as they were finitely many variables. Mathematically, this may be ill-defined.

    • The gauge theory could be anomalous.

    • There could be infinitely many gauge-for-gauge reducibility levels.

    • Rank conditions could jump.

References:

  1. I.A. Batalin & G.A. Vilkovisky, Gauge Algebra and Quantization, Phys. Lett. B 102 (1981) 27–31.

  2. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994.

  3. M. Henneaux, Lectures on the antifield-BRST formalism for gauge theories, Nucl. Phys. B Proc. Suppl. 18 (1990) 47.

  4. J. Gomis, J. Paris & S. Samuel, Antibracket, Antifields and Gauge-Theory Quantization, arXiv:hep-th/9412228.

Qmechanic
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  • thanks for the detailed answer. My doubt is in point 1, you said to identify physical D.O.F we can use light cone quantization, so can't we get that even by BV formalism as propagating D.O.F of the field would be n - n_0 + n_1 - ..... where n_0, n_1, ... are respective null space dimensions of generators at different level of reducible theory (as given in Gomis). – Bhavya Bhatt Feb 10 '21 at 14:14
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    Everything can in principle be deduced from the BRST formulation. – Qmechanic Feb 10 '21 at 14:34