Show that (σ·p)^2=|𝐩|^2.

Name: Problem 8, Chapter 5: Quarks and leptons: introductory course in modern particle physics
Uploaded: 2022-02-05T05:01:58-08:00
Duration: 1 min 2 s
Channel: Raj Bala
Description: Show that (σ·p)^2=|𝐩|^2.

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Key Concepts

Pauli Matrices

The Pauli matrices are a set of three 2x2 complex matrices that represent the spin operators for a spin-1/2 system. They possess key properties including being Hermitian and unitary, and they satisfy specific anticommutation relationships, such as {?_i, ?_j} = 2?_ij, which are critical in simplifying products of these matrices.

Anticommutation Relations

Anticommutation relations are fundamental in operator algebra, especially for matrices like the Pauli matrices. These relations, where the sum of the product of two operators in both orders equals twice a delta function, enable simplifications in expressions that involve the square of a linear combination of operators, such as (?·p)^2.

Operator Dot Product

The operator dot product, in cases like ?·p, represents a scalar obtained by summing the products of corresponding components from two vectors, one of operators (the Pauli matrices) and one of numbers or other operators (momentum components). Understanding how to manipulate this dot product using the underlying algebraic rules is key to demonstrating identities such as (?·p)^2 = |p|^2.

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