Notation clarification: SU(N) group integration

In summary, the equation 15.79 in Kaku's Quantum Field Theory book defines det( {\delta \over \delta J}) W(J) as the determinant of the matrix whose i, j'th element is \deltaW\deltaJij\deltaW\deltaJij. This determinant is calculated using the formula \det (M) = \frac{1}{n!} \epsilon_{i_{1} \cdots i_{n}} \ \epsilon_{j_{1} \cdots j_{n}} \ M_{i_{1}j_{1}} \ \cdots \ M_{i_{n}j_{n}} and by integrating over SU(n). This results in the equation
  • #1
paralleltransport
131
96
Homework Statement
This is not a homework problem.
Relevant Equations
U is a matrix element of SU(N). dU is the haar measure (left invariant measure) on the SU(N) lie group.
Hello,

I would like help to clarify what det( {\delta \over \delta J}) W(J) (equation 15.79) actually means, and why it returns a number (and not a matrix). This comes from the following problem statement (Kaku, Quantum Field Theory, a Modern Introduction)
1640643097208.png


Naively, one would define det ({\delta \over \delta J}) W(J) to be the determinant of the matrix whose i, j'th element is
δWδJijδWδJij
 
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  • #2
paralleltransport said:
I would like help to clarify what [itex]\det(\frac{\delta }{\delta J}) W(J)[/itex].
The determinant of any [itex]n \times n[/itex] matrix (such as [itex]u[/itex] and [itex]\frac{\delta}{\delta J}[/itex]) is given by [tex]\det (M) = \frac{1}{n!} \epsilon_{i_{1} \cdots i_{n}} \ \epsilon_{j_{1} \cdots j_{n}} \ M_{i_{1}j_{1}} \ \cdots \ M_{i_{n}j_{n}} .[/tex] Now take [itex]M = u \in SU(n)[/itex] and integrate over [itex]SU(n)[/itex]: since [itex]\det (u) = 1[/itex] and the measure is normalized [itex]\int_{SU(n)} d \mu (u) = 1[/itex], you get [tex]1 = \frac{1}{n!} \epsilon_{i_{1} \cdots i_{n}} \ \epsilon_{j_{1} \cdots j_{n}} \int_{SU(n)} d \mu (u) \ u_{i_{1}j_{1}} \ \cdots \ u_{i_{n}j_{n}} ,[/tex] or [tex]1 = \left( \frac{1}{n!} \epsilon_{i_{1} \ \cdots \ i_{n}} \ \epsilon_{j_{1} \ \cdots \ j_{n}} \frac{\delta}{\delta J_{i_{1}j_{1}}} \ \cdots \frac{\delta}{\delta J_{i_{n}j_{n}}} \right) W(J) \equiv \mbox{det}(\frac{\delta}{\delta J}) W(J) .[/tex]
 
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Likes paralleltransport
  • #3
OK that clears it up thanks!
 

Related to Notation clarification: SU(N) group integration

1. What is the SU(N) group?

The SU(N) group, also known as the special unitary group, is a mathematical group that consists of all N×N unitary matrices with determinant equal to 1. It is a fundamental concept in the study of quantum mechanics and has applications in various fields such as particle physics, quantum field theory, and condensed matter physics.

2. What does the integration in SU(N) group integration mean?

In mathematics, integration is a way of summing up infinitely many small quantities to find the total. In the context of SU(N) group integration, it refers to the process of calculating the integral of a function over the group, which is a measure of the group's size and shape.

3. How is SU(N) group integration different from regular integration?

SU(N) group integration is a specialized form of integration that is specific to the SU(N) group. It involves integrating over the group's elements, which are matrices, rather than over real or complex numbers as in regular integration. Additionally, the integration measure used in SU(N) group integration is different from the standard Lebesgue measure used in regular integration.

4. What is the purpose of SU(N) group integration in physics?

In physics, SU(N) group integration is used to calculate physical quantities that are related to the symmetries of a system described by the SU(N) group. This includes quantities such as particle decay rates, scattering amplitudes, and correlation functions. It allows for a more efficient and elegant way of handling these calculations compared to other methods.

5. Are there any applications of SU(N) group integration outside of physics?

Yes, there are various applications of SU(N) group integration in mathematics and other fields. It has applications in the study of Lie groups and Lie algebras, which are important mathematical structures in areas such as differential geometry, topology, and representation theory. It also has applications in engineering, specifically in control theory and signal processing.

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