Solving Computing a Trace Properties: Any Help Appreciated

In summary, there are several properties of trace operations involving gamma matrices and square matrices. These properties include the fact that if a product of gamma matrices contains an odd number of matrices, the trace of this product is equal to zero. Additionally, the trace of the product of two gamma matrices is equal to four times the product of the matrices. For a product of four gamma matrices, the trace is equal to four times the difference between the product of the first two matrices and the product of the last two matrices, plus four times the sum of the products of the first and third matrices, and the second and fourth matrices, minus four times the product of the first and fourth matrices and the second and third matrices. Lastly, for square matrices A
  • #1
JD_PM
1,131
158
Homework Statement
Compute the following trace

##Tr \Big( Y(\not{\!p_1'}+m) \Big) \ \ (1)##

Where

##\not{\!A} := \gamma^{\alpha} A_{\alpha} \ \ (2)##

##Y:= 4 \not{\!f_1} \not{\!p} \not{\!f_1} + m[-16(pf_1)+16 f_1^2] + m^2 ( 4 \not{\!p} - 16 \not{\!f_1})+16 m^3 \ \ (3)##


Source: Second Edition QFT, Mandl & Shaw page 144.
Relevant Equations
Please see properties below
Properties

1) If ##(\gamma^{\alpha}\gamma^{\beta}...\gamma^{\mu}\gamma^{\nu})## contains an odd number of ##\gamma##-matrices

$$Tr(\gamma^{\alpha}\gamma^{\beta}...\gamma^{\mu}\gamma^{\nu})=0$$

2)

$$Tr(\not{\!A}\not{\!B})=4AB$$

3)

$$Tr(\not{\!A}\not{\!B}\not{\!C}\not{\!D})=4\Big( (AB)(CD)-(AC)(BD)+(AD)(BC) \Big)$$

4) Given ##A, B## to be square matrices

$$Tr(A+B)=Tr(A) + Tr(B)$$

Besides: I've assumed that the trace of a scalar is equal to itself. (i.e. ##Tr(m)=m##)
Applying such properties I get

$$Tr \Big( Y(\not{\!p_1'}+m) \Big) = 16 \Big( (f_1 p)(f_1 p') - f_1^2(pp') + f_1p'p p' \Big) + 16m^2p^2-64 m^2 f_1 p + 16m^2[-pf_1+f_1^2]+16m^4 \ \ (4)$$

The provided solution is

$$Tr \Big( Y(\not{\!p_1'}+m) \Big)=16 \Big( 2(f_1p)(f_1p') -f_1^2(pp')+m^2[-4(pf_1)+4f_1^2]+m^2 [(pp')-4(f_1 p')] +4m^4 \Big) \ \ (5)$$

My solution is not equivalent to ##(5)##. What am I missing?Any help is appreciated.

Thank you :smile:

EDIT: I was wrong assuming that ##Tr(m)=m##; Of course that traces aren't defined for scalars! 😅. It should be ##m \times Tr (1_n) = n m##, where ##1_n## is the n-identity matrix
 
Last edited:
Physics news on Phys.org

Related to Solving Computing a Trace Properties: Any Help Appreciated

1. What is the purpose of solving computing trace properties?

The purpose of solving computing trace properties is to analyze and understand the behavior of a computer program or system. By computing the trace, we can track the sequence of instructions executed and the values of variables at each step, which can help identify errors and optimize performance.

2. What are some common trace properties that can be solved?

Some common trace properties that can be solved include correctness (whether the program produces the expected output), termination (whether the program terminates), and complexity (how much time and resources the program requires to run).

3. What techniques are used to solve computing trace properties?

There are various techniques that can be used to solve computing trace properties, such as model checking, theorem proving, and static analysis. These techniques involve using mathematical and logical methods to analyze the program code and verify its properties.

4. What challenges may arise when solving computing trace properties?

One of the main challenges when solving computing trace properties is the complexity of modern computer systems and programs. As systems become more sophisticated, it becomes increasingly difficult to analyze and verify their behavior. Additionally, the use of third-party libraries and components can introduce unexpected behaviors and make it harder to track the program's execution.

5. How can solving computing trace properties benefit the development process?

Solving computing trace properties can benefit the development process by helping to identify and fix errors early on, which can save time and resources in the long run. It can also provide insights into the program's performance and potential areas for optimization. Additionally, solving trace properties can increase confidence in the correctness and reliability of the program, which is crucial in safety-critical systems.

Similar threads

  • High Energy, Nuclear, Particle Physics
Replies
1
Views
1K
Replies
1
Views
3K
  • High Energy, Nuclear, Particle Physics
Replies
15
Views
2K
  • High Energy, Nuclear, Particle Physics
Replies
4
Views
1K
  • High Energy, Nuclear, Particle Physics
Replies
13
Views
3K
  • Advanced Physics Homework Help
Replies
8
Views
7K
  • High Energy, Nuclear, Particle Physics
Replies
2
Views
2K
  • Set Theory, Logic, Probability, Statistics
Replies
6
Views
1K
Replies
5
Views
2K
  • Special and General Relativity
Replies
1
Views
829
Back
Top