I have been working in the properties of the large gauge(adsbygoogle = window.adsbygoogle || []).push({});

transformation of QCD in the temporal gauge and I have shown that

these satisfy U_{n}U_{m} and commutes with the translations where the

large gauge transformations U_n and U_m belongs to the homotopy

classes characterized by winding numbers n and m. I prove that by

showing that n(U_1U_2)=n(U_1)+n(U_2) where U₁=U(_{n₁}) and U₂=U(_{n₂})

give representatives U_{n₁} and U_{n₂} (we have that U_{n₁} and U_{n₂}

are large gauge transformations) in each homotopy classes

characterized by winding numbers n₁=n(U₁) and n₂=n(U₂) and

n(U₁U₂)=n(U₁)+n(U₂)=n₁+n₂ is the winding number which characterizes

the homotopy classes of U₁U₂. For the winding number I have used the

expression n=(1/(24pi²))∫d³xepsilon^{ijk}Tr[U⁻¹∂_{i}UU⁻¹∂_{j}UU⁻¹∂_{k}U].

I have proved the large gauge transformations in QCD in the temporal

gauge commutes with the translations by showing that the winding

number n doesnot change when the translation

U(a)U(_{n})U⁻¹(a)=U(_{n}^{a}) is implemented under U(_{n}) where the

large trasformation U(_{n}) gives a only representative U_{n} in each

homotopy class characterized by a winding number n=n(U). It is correct

to use these argument to say that in the Nakanishi Lautrup gauge the

large gauge transformations of QCD have the same properties that in

the temporal gauge?. The expression

n=(1/(24²))∫d³x^{ijk}Tr[U⁻¹∂_{i}UU⁻¹∂_{j}UU⁻¹∂_{k}U] for the winding

number also holds in the Nakanishi Lautrup gauge?.

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# Large gauge transformations of QCD in the temporal and in the Nakanishi Lautrup gauge

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