# Higgs mechanism from a deformation quantization point of view

Is it possible to describe the Higgs mechanism from a deformation quantization point of view? How would one do it? Are there aspects of the Higgs mechanism and Higgs particle which one may see clearer from such a point of view?

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Some motivation for the question and your ideas about what kind of connection between these concepts you would want to see would be nice. – Igor Khavkine Sep 28 '12 at 13:43
The motivation is just out of curiosity. In recent times the possible discovery of the Higgs particle was a big topic in newspapers etc. Since I am interested in deformation quantization I just wanted to know if there is any connection between this approach to quantization and the Higgs mechanism especially if the deformation quantization approach could give a mathematically and conceptually clearer picture of the Higgs mechanism... – student Sep 28 '12 at 21:45

The Standard Model of particle physics (which includes the Higgs as one of its matter fields, and hence constitutes a description of Higgs particle and its interactions with other elementary particles) is a special case of a quantum field theory. One can think of a quantum field theory as the quantization of a classical field theory. A classical field theory is described by a Lagrangian density and the quantization procedure takes a Lagrangian as input, and producing a quantum field theory as output. Deformation quantization is a way of describing this quantization procedure. Thus, deformation quantization can be thought of as applied to any field theory (with the Standard Model just a special case) in the process of quantization. That's about as tight as one can draw a connection between deformation quantization and the Higgs particle itself.

Now, the phrase "the Higgs mechanism" that you used, has some technical meanings that are distinct from just describing the Higgs particle, which is what I discussed in the first paragraph. The Higgs mechanism usually refers to the fact that the low energy excitations about the symmetry broken vacuum of the Standard Model have non-zero masses, while the elementary fields have no explicit mass terms in the Lagrangian. Thus, the Higgs mechanism has all to do with the structure of the Standard Model Lagrangian and not much to do with the field quantization procedure itself (which takes any Lagrangian as input).

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The Higgs mechanism in the Standard Model doesn't have much to do with deformation quantization as other people have explained. However there is a version of the Higgs mechanism in string theory which involves stable D-branes arising via the Higgs mechanism from unstable D-branes. This is very hard to study directly in string field theory where one has to resort to approximate numerical techniques, but if one deforms the theory by turning on a B field as discussed in Seiberg and Witten, hep-th/9908142 one obtains a noncommutative version and in the "large B limit" one can obtain exact results. For details see hep-th/0005031 and references therein.

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I think that everybody who just starts looking to "quantization related things" will be at first moment be overhelmed by various words "deformation quantization" "geometric quantization" "secondary quantization" "quantum groups" ...... and taming this zoo will at least take some time. Probably this is a part of "motivation" for the question. Let me give some comments on this.

1) I do not think deformation quantization is directly helpful for Higgs mechanism, as well as for most other questions of most other realistic quantum field theories.

2) The reason is that realistic quantum field are usually written in "canonical coordinates" e.g. [p,q]=1 so deformation quantization coincide with "canonical quantization", and in this case it brings nothing new to what physitits knew starting from the times of Heisenberg, Pauli, Dirac ~1930...

Well, there are subtleties related to gauge symmetries, but reasonable way to treat them is various cohomological BRST, BV methods...

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The Higgs field basically satisfies a nonlinear Klein-Gordon (KG) equation.The nonlinearity stems from the existence of a nonlinear potential energy term (interaction) depending on the field but not on its derivatives. The "classical" Higgs mechanism takes place when the energy functional corresponding to this equation acquires a global minimum for a nonvanishing value of the field.

Although I am not aware of any work treating the Higgs mechanism within the framework of deformation quantization, I want to refer you to the following work by Giuseppe Dito who defined a deformation quantization and a corresponding star product of interacting Klein-Gordon fields (thus can be applied in principle to the Higgs mechanism problem). This star product acts on the free asymptotic fields corresponding to the solutions of the nonlinear KG equation.

An important feature of this star product is that it has a built-in renormalization mechanism, as the divergent terms that can be encountered in its application can be attributed to either coboundaries in its Hochschild cohomology or cocycles that can be removed by a modification of only lower order terms in $\hbar$. In either way a finite star product at each order of $\hbar$ can be found.

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Thanks for pointing out Dito's paper I didn't know it and was not aware that it is possible have star products which has a "built-in renormalization mechanism"! – student Oct 24 '12 at 12:14