Proving Poincare Algebra Using Differential Expression of Generator

In summary: The ##x##'s are c-numbers and commute with everything. The ##p##'s are going to be operators, so they need to be really careful, and put the minus sign in the definition of the operator.In summary, using differential expressions for the generator, we can verify the commutator expression ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in the Poincare group. However, one must be careful with sign conventions for ##P_\rho## and ##J_{\mu \nu}##, as the definition of ##J
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Homework Statement
Using differential expressions for the generator to verify the commutator expression in Poincare group
Relevant Equations
Definition for the diffrential expressions of the generators are given below
Using differential expressions for the generator, verify the commutator expression for ##[J_{\mu\nu},P_{\rho}]=i(\eta_{\mu\rho}P_{\nu}-\eta_{\nu\rho}P_{\mu})## in Poincare group

Generator of translation: ##P_{\rho}=-i\partial_{\rho}##
Generator of rotation: ##J_{\mu\nu}=i(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})##

Here is my working, I operate the commutator on a vector ##x^j##:
##[J_{\mu\nu},P_{\rho}]x^j##
##=(J_{\mu\nu}P_{\rho}-P_{\rho}J_{\mu\nu})x^j##
##=0-iP_{\rho}(x_{\mu}\partial_{\nu}-x_{\nu}\partial{_\mu})x^j##
##=-[(\partial_{\rho}x_{\mu})(\partial_{\nu}x^j)-(\partial_{\rho}x_{\nu})(\partial_{\mu}x^j)]##
##=-[\eta_{\rho\mu}(\partial_{\nu}x^j)-\eta_{\rho\nu}(\partial_{\mu}x^j)]##
##=-i(\eta_{\rho\mu}P_{\nu}x^j-\eta_{\rho\nu}P_{\mu}x^j)##
##=-i(\eta_{\rho\mu}P_{\nu}-\eta_{\rho\nu}P_{\mu})x^j##

My answer had one extra negative sign on it. Where did it go wrong?
 
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crime9894 said:
My answer had one extra negative sign on it. Where did it go wrong?
Your work looks correct to me. One has to be careful with sign conventions for ##P_\rho## and ##J_{\mu \nu}##. For example, https://www.physik.uni-bielefeld.de/~borghini/Teaching/Symmetries/02_02.pdf they get the answer that you were asked to get, but they define ##J_{\mu \nu}## with the opposite sign.
 
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  • #3
And it should have the opposite sign, because ##J_{\mu \nu}=x_{\mu} p_{\nu}-x_{\nu} p_{\mu}## and then you just make the symbols operators.
 

1. What is the Poincare Algebra?

The Poincare Algebra is a mathematical structure that describes the symmetries of space and time in physics. It consists of a set of generators, such as translations and rotations, and their associated commutation relations.

2. How is the Poincare Algebra proven using differential expression of generators?

The Poincare Algebra can be proven using differential expression of generators by showing that the generators satisfy the same commutation relations as the algebra. This involves using the differential operators to express the generators and then manipulating them to show that they satisfy the desired commutation relations.

3. What is the significance of proving the Poincare Algebra?

Proving the Poincare Algebra is significant because it provides a mathematical foundation for understanding the symmetries of spacetime in physics. It also allows for the development of theories and models that accurately describe the behavior of physical systems.

4. Are there any challenges in proving the Poincare Algebra using differential expression of generators?

Yes, there can be challenges in proving the Poincare Algebra using differential expression of generators. This may include complex mathematical manipulations and the need for a deep understanding of both algebra and differential equations.

5. How is the proof of the Poincare Algebra using differential expression of generators used in physics?

The proof of the Poincare Algebra using differential expression of generators is used in physics to understand the symmetries of spacetime and to develop theories and models that accurately describe physical systems. It is also used in quantum field theory and other areas of physics to study the behavior of particles and their interactions.

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