# Proving Feynman Slash Identity: 2a\cdot b

• nicksauce
In summary, the conversation is about proving the relation \displaystyle{\not} a \displaystyle{\not} b + \displaystyle{\not} b \displaystyle{\not} a = 2a\cdot b using the given relation \{\gamma^{\mu},\gamma^{\nu}\} = 2g^{\mu\nu}. The person is attempting to work backwards and use the fact that a_\mu and b_\nu are ordinary numbers that commute with everything.
nicksauce
Homework Helper

## Homework Statement

I am trying to prove that $$\displaystyle{\not} a \displaystyle{\not} b + \displaystyle{\not} b \displaystyle{\not} a = 2a\cdot b$$ using the relation $$\{\gamma^{\mu},\gamma^{\nu}\} = 2g^{\mu\nu}$$

## The Attempt at a Solution

If I work backwards,
$$2a\cdot b = 2a_{\mu} g^{\mu \nu} b_{\nu} = a_{\mu}(\gamma^{\mu}\gamma^{\nu})b_{\nu} + a_{\mu}(\gamma^{\nu}\gamma^{\mu})b_{\nu}$$

The first term is $$\displaystyle{\not} a \displaystyle{\not} b$$ but the second term doesn't seem to look like $$\displaystyle{\not} b \displaystyle{\not} a$$. Am I missing something here?

You're missing the fact that $a_\mu$ and $b_\nu$ are ordinary numbers, and so commute with everything.

## 1. What is the Feynman slash identity?

The Feynman slash identity is a mathematical identity used in quantum field theory, named after the physicist Richard Feynman. It is a way to represent the product of two four-vectors in terms of the four-vector dot product and the Dirac gamma matrices.

## 2. What is the purpose of proving the Feynman slash identity?

The Feynman slash identity is a fundamental equation in quantum field theory, and proving it ensures that the equation is mathematically valid and can be used as a tool in calculations and experiments.

## 3. How is the Feynman slash identity proven?

The Feynman slash identity is proven using algebraic manipulations and properties of the Dirac gamma matrices. It involves expanding the product of two four-vectors using their components and simplifying the terms using the properties of the gamma matrices.

## 4. Are there any applications of the Feynman slash identity?

Yes, the Feynman slash identity has numerous applications in quantum field theory, particularly in calculations involving scattering amplitudes and Feynman diagrams. It is also used in particle physics to describe the interactions of subatomic particles.

## 5. What are the implications of not proving the Feynman slash identity?

If the Feynman slash identity is not proven, it cannot be considered a valid mathematical equation and cannot be used in calculations or experiments. This could hinder progress in the field of quantum field theory and particle physics, as the identity is a crucial tool in understanding and predicting the behavior of subatomic particles.

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