How does non-commutativity emerge from path integral?

In summary, the non-commutative nature of Quantum Mechanics in path integral formulation may not be obvious at first glance. However, by constructing the path integral from the Hamiltonian and properly ordering the operators, the commutation relations are naturally invoked. This can be seen by breaking the path integral into smaller pieces and inserting the resolution of identity with the appropriate states. This ultimately leads to the identification of the Lagrangian and the commutation relation is then demonstrated.
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  • #2
Consider constructing the path integral from the Hamiltonian. You need the Hamiltonian to be in ordered a particular way --- if you're using coherent states, then normal ordered (or anti-), or if you're using q and p, then again some definite ordering of polynomial terms. You sandwich exp(-Ht) between the initial and final states, then break this up into N small pieces, inserting resolution of the identity with the states |x> and |p> or whatever else you need, remembering that you want the correct states to hit the right things in the Hamiltonian. This then bring out the term which looks like $p \partial_t q$ in the final expression, and we identify the whole shebang along with -H as the lagrangian.
 
  • #3
I see, when trying to order the operators the commutation relation is naturally invoked. But I don't see how to use the argument in the link I cited, to show the non-commutativity.
 

1. What is non-commutativity in the context of path integral?

Non-commutativity refers to the property of two operators not being able to be multiplied in any order and still produce the same result. In the context of path integral, it means that the order in which the integration variables are arranged can affect the final result.

2. How does non-commutativity emerge from path integral?

Non-commutativity in path integral arises from the fact that the path integral is a sum over all possible paths and the order in which these paths are summed can affect the result. This is because the integration variables in the path integral are operators that do not commute with each other.

3. Can you provide an example of non-commutativity in path integral?

One example of non-commutativity in path integral is in quantum mechanics, where the position and momentum operators do not commute. This means that the order in which the path integral is calculated can affect the probability of a particle being in a certain position at a certain time.

4. How does non-commutativity affect our understanding of physical systems?

Non-commutativity in path integral has important implications for our understanding of physical systems, particularly in quantum mechanics. It means that the order in which operators are applied can affect the outcome, leading to phenomena such as quantum entanglement and uncertainty.

5. Are there any real-world applications of non-commutativity in path integral?

Yes, non-commutativity in path integral has various applications in fields such as quantum field theory, statistical mechanics, and condensed matter physics. It is also used in the development of quantum algorithms for quantum computing.

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