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Winding number in 4D & SU(2) group

1

$begingroup$

In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such that

$$hatx = (sinchi sinpsi cosphi, sinchi sinpsi sinphi, sinchi cospsi, coschi), quad sum_mu hatx_mu hatx_mu = 1$$

$n$ is given by

$$
n = -frac124pi^2int_0^pi dchiint_0^pi dpsi int_0^2pi dphi epsilon^alphabetagammatr(Upartial_alpha U^dagger) (Upartial_beta U^dagger) (Upartial_gamma U^dagger), quad epsilon^chipsiphi = +1
$$

Where $U$ is only dependent on $hatx$, belongs to $SU(2)$ and has associated the winding number $n$. $tr$ represents the trace.

But suddenly Srednicki says that you can write $n$ as an integral over the surface of this 4-dimensional space of the form

$$
n = frac124pi^2int dS_mu epsilon^munusigmatautr(Upartial_nu U^dagger) (Upartial_sigma U^dagger) (Upartial_tau U^dagger), quad partial_nu = partial/partial x^nu rm and so on
$$

I don't understand how you can go from one expression of $n$ to the other.

integration lie-groups topological-groups multiple-integral winding-number

share|cite|improve this question

edited Apr 8 at 7:49

Vicky

asked Apr 8 at 5:51

VickyVicky

2627

$endgroup$

add a comment | 

1

$begingroup$

In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such that

$$hatx = (sinchi sinpsi cosphi, sinchi sinpsi sinphi, sinchi cospsi, coschi), quad sum_mu hatx_mu hatx_mu = 1$$

$n$ is given by

$$
n = -frac124pi^2int_0^pi dchiint_0^pi dpsi int_0^2pi dphi epsilon^alphabetagammatr(Upartial_alpha U^dagger) (Upartial_beta U^dagger) (Upartial_gamma U^dagger), quad epsilon^chipsiphi = +1
$$

Where $U$ is only dependent on $hatx$, belongs to $SU(2)$ and has associated the winding number $n$. $tr$ represents the trace.

But suddenly Srednicki says that you can write $n$ as an integral over the surface of this 4-dimensional space of the form

$$
n = frac124pi^2int dS_mu epsilon^munusigmatautr(Upartial_nu U^dagger) (Upartial_sigma U^dagger) (Upartial_tau U^dagger), quad partial_nu = partial/partial x^nu rm and so on
$$

I don't understand how you can go from one expression of $n$ to the other.

integration lie-groups topological-groups multiple-integral winding-number

share|cite|improve this question

edited Apr 8 at 7:49

Vicky

asked Apr 8 at 5:51

VickyVicky

2627

$endgroup$

add a comment | 

$begingroup$

In the book 'Quantum field theory' by Mark Srednicki (chapter 93, pages 575-576) in order to compute winding number, $n$, in a 4-dimensional space with coordinates $x = (x_1, x_2, x_3, x_4)$ and such that

$$hatx = (sinchi sinpsi cosphi, sinchi sinpsi sinphi, sinchi cospsi, coschi), quad sum_mu hatx_mu hatx_mu = 1$$

$n$ is given by

$$
n = -frac124pi^2int_0^pi dchiint_0^pi dpsi int_0^2pi dphi epsilon^alphabetagammatr(Upartial_alpha U^dagger) (Upartial_beta U^dagger) (Upartial_gamma U^dagger), quad epsilon^chipsiphi = +1
$$

Where $U$ is only dependent on $hatx$, belongs to $SU(2)$ and has associated the winding number $n$. $tr$ represents the trace.

But suddenly Srednicki says that you can write $n$ as an integral over the surface of this 4-dimensional space of the form

$$
n = frac124pi^2int dS_mu epsilon^munusigmatautr(Upartial_nu U^dagger) (Upartial_sigma U^dagger) (Upartial_tau U^dagger), quad partial_nu = partial/partial x^nu rm and so on
$$

I don't understand how you can go from one expression of $n$ to the other.

integration lie-groups topological-groups multiple-integral winding-number

share|cite|improve this question

edited Apr 8 at 7:49

Vicky

asked Apr 8 at 5:51

VickyVicky

2627

$endgroup$

share|cite|improve this question

edited Apr 8 at 7:49

Vicky

asked Apr 8 at 5:51

VickyVicky

2627

share|cite|improve this question

edited Apr 8 at 7:49

Vicky

asked Apr 8 at 5:51

VickyVicky

2627

asked Apr 8 at 5:51

VickyVicky

2627

asked Apr 8 at 5:51

VickyVicky

2627