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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

User Tagabek
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Final Answer:

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.

User Samaria
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