# Working out a Second Class Constraint in Gory detail?

• pqnelson
In summary, the conversation is about a person who has encountered a second class constraint while working with equations and is looking for a good introduction to the subject. They have made the conjugate momentum a function of position, resulting in a non-zero value for [C^i, C^j]. They are seeking help in understanding what this value is, but are unable to due to their limited knowledge of quantum field theory. The conversation ends with the person stating that they have figured out the issue themselves.
pqnelson
Hello,

I've been fooling around with some equations, and I've managed to land a second class constraint (I don't know whether to laugh or cry). Well, I was wondering is there any good introduction to the subject?

The problem I have is that I made the conjugate momentum a function of position, so I have the constraint:
C^i = \pi^i - p^i = 0
and thus
[C^i, C^j] \neq 0
and I would like to know what exactly [C^i, C^j] is equal to; however, I've never really learned quantum field theory "traditionally" and so I'm completely stuck.

Any help would be greatly appreciated!

If i don't know who $\pi^{i}$ and $p^{i}$ are, i can't really help you.

Daniel.

P.S. It's enough to post the lagrangian (density) you started with.

Well, I think I've got it figured out now, thanks anyways.

## 1. What is a second class constraint?

A second class constraint is a mathematical condition that arises in the process of quantizing a classical physical system. It refers to a constraint that does not have a unique solution and requires further analysis to determine its precise form.

## 2. Why is working out a second class constraint important?

Working out a second class constraint is important because it allows for a more accurate and complete understanding of a physical system. By identifying and quantifying these constraints, we can better describe the behavior of the system and make more accurate predictions about its future behavior.

## 3. What is the process for working out a second class constraint?

The process for working out a second class constraint involves first identifying the constraint in the classical system. Next, the constraint is quantized and expressed in terms of quantum operators. Finally, the constraint is solved using mathematical techniques such as Dirac brackets or the Faddeev-Jackiw method.

## 4. Can second class constraints be ignored?

No, second class constraints cannot be ignored. They are a fundamental part of the quantization process and play a crucial role in accurately describing the behavior of a physical system. If ignored, the resulting quantum theory would be incomplete and may lead to incorrect predictions.

## 5. Are there any real-world applications of working out second class constraints?

Yes, there are many real-world applications of working out second class constraints. They have been used in various fields such as quantum mechanics, general relativity, and particle physics to accurately describe and predict the behavior of complex physical systems. They are also essential in the development of new technologies such as quantum computing and high-energy particle accelerators.

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