Can Noncommutative Operators Be Integrated in Quantum Mechanics?

[mhill]

since in QM we deal with noncommutative operators , for example

$$[A,B]=i\hbar 1$$

then my question is if we could define some kind of noncommutative integral

$$\int dA dB exp(-iAB)$$ here A and B would be operators

how can you define for an operator a 'measure' $$dA$$

 

[Avodyne]

In general one never integrates over operators in QM.

 

[denian]

, and how can you integrate over two noncommutative operators A and B?

In quantum mechanics, we often deal with noncommutative operators, meaning that the order in which they are applied matters. This is represented by the commutator [A,B], which is defined as the difference between the product of the operators A and B in different orders. This is a fundamental property of operators in quantum mechanics and is related to the uncertainty principle.

In the context of integration, we typically integrate over continuous variables, such as position or momentum. However, it is possible to extend this concept to operators. This is known as operator integration.

In the case of noncommutative operators, the traditional definition of integration does not apply. Instead, we can use the concept of a trace, which is a mathematical operation that maps an operator onto a scalar value. The trace of an operator can be thought of as a generalized integral, and it allows us to integrate over noncommutative operators.

In the example given, we are asked to consider the integral \int dA dB exp(-iAB), where A and B are operators. This type of integral can be defined using the trace operation, and it allows us to integrate over two noncommutative operators.

However, in order to define a 'measure' dA for an operator, we must first define a suitable space for the operators to live in. This is known as a Hilbert space, and it is a mathematical structure that allows us to perform operations on operators, such as integration.

Once we have a Hilbert space, we can then define a measure for operators, which allows us to integrate over them. This measure is known as the spectral measure, and it is essentially a way of assigning a weight to each possible outcome of an operator measurement.

Integrating over noncommutative operators is a powerful tool in quantum mechanics, as it allows us to calculate important quantities such as expectation values and probabilities. However, it requires a strong mathematical understanding of operator theory and Hilbert spaces. Overall, the concept of operator integration is essential in understanding the behavior of quantum systems and is a crucial tool for scientists in the field.