Question

Express each of the following equations in a form that does not involve isometries: a.σAσB = σBσA b.σBσA = σCσB c.σBσa = σbσB d.σbσA = σBσb e.σbσA = σAσb f.σbσa = σcσb

Answer 1

a.
`\( \sigma_A \sigma_B = \sigma_B \sigma_A \)` is satisfied when
`\( \sigma_A \)` and
`\( \sigma_B \)` commute, leading to the identity matrix.

b.
`\( \sigma_B \sigma_A = \sigma_C \sigma_B \)` implies
`\( \sigma_A \)` and
`\( \sigma_B \)` do not commute, and
`\( \sigma_C \)` can be expressed as
`\( \sigma_C = \sigma_A \sigma_B \sigma_A^(-1) \)`.

c.
`\( \sigma_B \sigma_a = \sigma_b \sigma_B \)` holds if
`\( \sigma_a \)` and
`\( \sigma_b \)` anticommute, resulting in
`\( \sigma_B \sigma_a = -\sigma_a \sigma_B \).`

Explanation:

In equation a,
`\( \sigma_A \sigma_B = \sigma_B \sigma_A \)` indicates that the Pauli matrices
`\( \sigma_A \) and \( \sigma_B \)` commute. The product of commuting matrices results in the identity matrix, yielding a simple and direct solution. This reflects a scenario where the order of operations does not affect the outcome.

In equation b,
`\( \sigma_B \sigma_A = \sigma_C \sigma_B \)`, the non-commutativity of
`\( \sigma_A \)` and
`\( \sigma_B \)` is implied. The presence of
`\( \sigma_C \)` suggests the introduction of an additional Pauli matrix. This equation is satisfied when
`\( \sigma_C \)` can be expressed as
`\( \sigma_C = \sigma_A \sigma_B \sigma_A^(-1) \)`, indicating a conjugation operation that compensates for the non-commutativity.

Equation c,
`\( \sigma_B \sigma_a = \sigma_b \sigma_B \)`, involves the anticommutation of
`\( \sigma_a \)` and
`\( \sigma_b \)`. Anticommutation implies that the product of the matrices is equivalent to the negation of the product in the opposite order. This relationship captures the essence of anti-symmetry in the Pauli algebra, where the interchange of operators introduces a sign change in the result.