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If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C
(B) ax ay az =[ax ay az B, B]?
(B) ax ay az =[ax ay az B, B]?
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If a and ẞare Dirac matrices, then which of the following is not corre...
Understanding Dirac Matrices
Dirac matrices are crucial in quantum mechanics and quantum field theory, particularly in the formulation of the Dirac equation. They are used to describe fermions, such as electrons.
Analysis of Each Option
- (A) \( \alpha_x = \frac{1}{2} [\alpha_x, \alpha_y] \)
This statement implies a relationship between Dirac matrices based on their commutation. However, Dirac matrices satisfy the Clifford algebra, which states that \( \{\alpha_i, \alpha_j\} = 2\delta_{ij}I \). Thus, this expression is not correct as it misrepresents the algebraic properties.
- (B) \( \alpha_x \alpha_y \alpha_z = [\alpha_x, \alpha_y] \alpha_z + \{B, B\} \)
This expression seems to suggest a combination of commutation and anti-commutation relations. However, the product of three Dirac matrices does not equal just the commutator of the first two matrices multiplied by the third. Therefore, this statement is also not accurate.
- (C) \( \text{Tr}(\bar{A} B) (\bar{A} C) = B \cdot C \)
This equation is a form of inner product in the context of quantum mechanics. The trace of a product of matrices is indeed related to the inner product of corresponding vectors. This statement holds true.
Conclusion
In summary, both options (A) and (B) contain inaccuracies regarding the properties of Dirac matrices. However, option (A) is particularly misleading as it misrepresents the fundamental algebra governing Dirac matrices. Therefore, the statement that is not correct is (A).
Dirac matrices are crucial in quantum mechanics and quantum field theory, particularly in the formulation of the Dirac equation. They are used to describe fermions, such as electrons.
Analysis of Each Option
- (A) \( \alpha_x = \frac{1}{2} [\alpha_x, \alpha_y] \)
This statement implies a relationship between Dirac matrices based on their commutation. However, Dirac matrices satisfy the Clifford algebra, which states that \( \{\alpha_i, \alpha_j\} = 2\delta_{ij}I \). Thus, this expression is not correct as it misrepresents the algebraic properties.
- (B) \( \alpha_x \alpha_y \alpha_z = [\alpha_x, \alpha_y] \alpha_z + \{B, B\} \)
This expression seems to suggest a combination of commutation and anti-commutation relations. However, the product of three Dirac matrices does not equal just the commutator of the first two matrices multiplied by the third. Therefore, this statement is also not accurate.
- (C) \( \text{Tr}(\bar{A} B) (\bar{A} C) = B \cdot C \)
This equation is a form of inner product in the context of quantum mechanics. The trace of a product of matrices is indeed related to the inner product of corresponding vectors. This statement holds true.
Conclusion
In summary, both options (A) and (B) contain inaccuracies regarding the properties of Dirac matrices. However, option (A) is particularly misleading as it misrepresents the fundamental algebra governing Dirac matrices. Therefore, the statement that is not correct is (A).
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If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]?
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If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]?.
If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If a and ẞare Dirac matrices, then which of the following is not correct? 1 (A) ax = 1/2 [axay, ay] (C) Tr(ā. B) (ā. C) = B. C (B) ax ay az =[ax ay az B, B]?.
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