Functional Integral in Free-Field Theory: Understanding the Derivation

In summary, the conversation discusses the use of functional integrals in free-field theory and the ability to solve them exactly. The main focus is on a specific step in the derivation where a unitary transformation is used to change integration variables. The claim is made that this transformation is unitary, and therefore the product of the integration variables can be written as the product of the real and imaginary parts. However, the individual's understanding of this claim is unclear and they request further explanation. The conversation also briefly mentions the use of the Jacobian determinant in this transformation and the importance of integrating only over certain wavevectors.
  • #1
jdstokes
523
1
In free-field theory, the functional integral

[itex]\int \mathcal{D}\varphi \exp\left(i \frac{1}{2} \int d^4 x (\partial_\mu \varphi \partial^\mu \varphi - m^2 \varphi^2)\right)[/itex]

can be done exactly (see e.g., Peskin and Schroeder p. 285).

I'm tyring to understand the step in their derivation where they change integration variables from the field [itex]d\varphi(x)[/itex], to the real and imaginary parts [itex]d\Re[\varphi(x)],d\Im[\varphi(x)][/itex]. They claim that since the transformation is unitary, they have

[itex]\prod_i d\varphi(x_i) = \prod_i d\Re[\varphi(x_i)]d\Im[\varphi(x_i)][/itex].

I don't understand this claim. Suppose the unitary xfm relating [itex]x_i[/itex] to [itex]X_i[/itex] is [itex]U[/itex]. Then inEinstein notation,

[itex]dx_i = U_{ij} dX_j [/itex].

Hence

[itex]\prod_i dx_i = \prod_i U_{ij} dX_j = (U_{1i}U_{2j}U_{3k}\cdots)(dX_i dX_j dX_k \cdots)[/itex].

Thus P&S's claim amounts to the assertion that

[itex]\prod_{n=1} U_{n ,i_n} = \prod_{n=1}\delta_{n, i_n}[/itex].

I don't understand this?

Any help would be appreciated.
 
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  • #2
First of all, it's just a definition. The measure for ordinary integration over a complex variable [itex]z=x+iy[/itex] is defined to be [itex]dx\,dy[/itex].

More generally, a change of variable involves the determinant of the jacobian matrix of the transformation.
 
  • #3
That's interesting, I desperately need to take a course in complex analysis.

I also forgot that the change of variables involves the Jacobian determinant, which is unity for a unitary matrix.

I still don't understand why P&S go through a long argument involving integrating only over the wavevectors k such that [itex]k^0 > 0[/itex]?
 

1. What is a functional integral in free-field theory?

A functional integral is a mathematical tool used in quantum field theory to calculate the probability amplitude for a system to evolve from an initial state to a final state. In free-field theory, it is used to describe the behavior of a field in the absence of any external forces or interactions.

2. How is the functional integral derived in free-field theory?

The functional integral in free-field theory is derived using the path integral formulation, which converts the summation over all possible field configurations into an integral over all possible paths. This allows for a more intuitive and elegant way of calculating the probability amplitude.

3. What is the significance of the functional integral in free-field theory?

The functional integral is a powerful tool in free-field theory as it allows for the calculation of physical quantities such as correlation functions and scattering amplitudes. It also provides a bridge between classical and quantum field theories, making it an important concept in theoretical physics.

4. Are there any limitations to using the functional integral in free-field theory?

While the functional integral is a useful tool, it can be difficult to calculate in some cases, particularly when dealing with non-perturbative effects. Additionally, it is only applicable for systems with a large number of degrees of freedom, making it less useful for systems with few particles.

5. How does the functional integral relate to other concepts in physics?

The functional integral in free-field theory is closely related to other mathematical concepts such as Feynman diagrams and path integrals. It is also connected to the concept of symmetry in physics, as the functional integral can be used to calculate the effects of symmetries on a system.

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