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

1. Dec 6, 2007

### aecarcamoh2005

I have been working in the properties of the large gauge
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?.