Canonical vs. path integral quantization

In summary, the conversation discusses the relationship between canonical and path integral quantization, and how the path integral formulation does not require the introduction of commutation relations between fields. However, the quantum nature arises in the computation of amplitudes through the use of a complex Boltzmann factor. The clarity of this explanation in textbooks may vary.
  • #1
earth2
86
0
Hey folks,

i have a question concerning canonical and path integral quantization.
From what I have understood so far, these two techniques are different and independent but equivalent.

My problem is that I don't really see where the quantum character enters in the path intregral formulation.

I mean, sure, it is based on bra and kets and all that stuff but don't the appearing fields in a path integral have to fullfil commutation relations, too?

So, before i can build a path integral, i have to make sure that the appearing fields fullfil certain (anti-)commutators, right? But doesn't this mean that the path integral formulation needs a canical formulation first?

Thanks for your answers!
Cheers,
earth2
 
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  • #2
No, everything that is put into the path integral is classical. That's why the formalism is so appealing: you don't have those nasty non-commuting operators. The path integral formalism does make use of the Hilbert space of the theory, but you do not need to introduce commutation relations between the fields.

The quantum nature arises in the computation of amplitudes: you sum over all "possible" contributions to an amplitude, and weighs these with a complex Boltzmann factor [tex]e^{i S}[/tex]. The fact that this factor is complex is what causes the quantum mechanical nature of the amplitudes.
 
  • #3
Thank you so much! Why isn't it written this clearly in textbooks? :)
 
  • #4
It is, for instance try Feynman and Hibbs
 

Related to Canonical vs. path integral quantization

1. What is the difference between canonical and path integral quantization?

Canonical quantization is a method used in quantum mechanics to convert a classical system into a quantum system. It involves promoting classical variables, such as position and momentum, to operators that obey the rules of quantum mechanics. On the other hand, path integral quantization is a different approach where the quantum state is described as a sum over all possible paths taken by a particle.

2. Which method is more commonly used in modern physics?

Both canonical and path integral quantization are used in modern physics, but path integral quantization is more commonly used in theoretical physics and quantum field theory. Canonical quantization is often used in more traditional approaches to quantum mechanics.

3. How do canonical and path integral quantization differ in their treatment of time evolution?

In canonical quantization, time evolution is described by the Schrödinger equation, which evolves the state of a quantum system forward in time. In path integral quantization, time evolution is described by summing over all possible paths taken by a particle, including both forward and backward in time.

4. Can both canonical and path integral quantization be used for systems with infinite degrees of freedom?

Yes, both methods can be used for systems with infinite degrees of freedom. However, path integral quantization is often preferred for these systems because it does not require solving differential equations, making it easier to apply to more complex systems.

5. Are there any limitations to using path integral quantization?

One limitation of path integral quantization is that it is difficult to apply to systems with constraints, such as gauge symmetries. In these cases, canonical quantization may be a better approach. Additionally, path integral quantization can be computationally intensive and may not always provide a unique solution.

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