In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz. The Fierz identities are also sometimes called the Fierz–Pauli–Kofink identities, as Pauli and Kofink described a general mechanism for producing such identities. When working in 4 spacetime dimensions the bivector may be decomposed in terms of the Dirac matrices that span the space: . The coefficients are where
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| - Fierz identity (en)
- 피어츠 항등식 (ko)
- Тождества Фирца (ru)
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| - 피어츠 항등식(Fierz identity)이란 두 스피너 쌍선형 형식의 곱을 다른 스피너 쌍선형 형식 곱의 선형 결합으로 나타내는 항등식이다. 스위스의 물리학자 마르쿠스 에두아르트 피어츠(Markus Eduard Fierz)가 도입하였다. 양자장론에서 스피너로 나타내어지는 페르미온을 다룰 때 쓴다. (ko)
- Тождества Фирца — тождества линейной алгебры, связывающие различные выражения в виде произведений матриц Паули, матриц Гелл-Манна и матриц Дирака, различающиеся между собой перестановкой индексов. Используются в теоретической физике. (ru)
- In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz. The Fierz identities are also sometimes called the Fierz–Pauli–Kofink identities, as Pauli and Kofink described a general mechanism for producing such identities. When working in 4 spacetime dimensions the bivector may be decomposed in terms of the Dirac matrices that span the space: . The coefficients are where (en)
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| - In theoretical physics, a Fierz identity is an identity that allows one to rewrite bilinears of the product of two spinors as a linear combination of products of the bilinears of the individual spinors. It is named after Swiss physicist Markus Fierz. The Fierz identities are also sometimes called the Fierz–Pauli–Kofink identities, as Pauli and Kofink described a general mechanism for producing such identities. There is a version of the Fierz identities for Dirac spinors and there is another version for Weyl spinors. And there are versions for other dimensions besides 3+1 dimensions. Spinor bilinears in arbitrary dimensions are elements of a Clifford algebra; the Fierz identities can be obtained by expressing the Clifford algebra as a quotient of the exterior algebra. When working in 4 spacetime dimensions the bivector may be decomposed in terms of the Dirac matrices that span the space: . The coefficients are and are usually determined by using the orthogonality of the basis under the trace operation. By sandwiching the above decomposition between the desired gamma structures, the identities for the contraction of two Dirac bilinears of the same type can be written with coefficients according to the following table. where The table is symmetric with respect to reflection across the central element. The signs in the table correspond to the case of commuting spinors, otherwise, as is the case of fermions in physics, all coefficients change signs. For example, under the assumption of commuting spinors, the V × V product can be expanded as, Combinations of bilinears corresponding to the eigenvectors of the transpose matrix transform to the same combinations with eigenvalues ±1. For example, again for commuting spinors, V×V + A×A, Simplifications arise when the spinors considered are Majorana spinors, or chiral fermions, as then some terms in the expansion can vanish from symmetry reasons.For example, for anticommuting spinors this time, it readily follows from the above that (en)
- 피어츠 항등식(Fierz identity)이란 두 스피너 쌍선형 형식의 곱을 다른 스피너 쌍선형 형식 곱의 선형 결합으로 나타내는 항등식이다. 스위스의 물리학자 마르쿠스 에두아르트 피어츠(Markus Eduard Fierz)가 도입하였다. 양자장론에서 스피너로 나타내어지는 페르미온을 다룰 때 쓴다. (ko)
- Тождества Фирца — тождества линейной алгебры, связывающие различные выражения в виде произведений матриц Паули, матриц Гелл-Манна и матриц Дирака, различающиеся между собой перестановкой индексов. Используются в теоретической физике. (ru)
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