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Mathematics of Quantization and Quantum Fields by Christian Gérard, Jan Derezinski

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7

Anti-symmetric calculus

In almost every respect there exists a strong analogy between symmetric and anti-symmetric tensors, between bosons and fermions. It is often convenient to stress this analogy in terminology and notation.

Symmetric tensors over a vector space can be treated as polynomial functions on its dual. Such functions can be multiplied, differentiated and integrated, and we can change their variables.

There exists a similar language in the case of anti-symmetric tensors. It has been developed mostly by Berezin, hence it is sometimes called the Berezin calculus. It is often used by physicists, because it allows them to treat bosons and fermions within the same formalism.

Anti-symmetric calculus has a great appeal – it often allows ...

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