Griffiths Introduction to Elementary Particles · Problem 3.8

Symmetries — David J. Griffiths

Physics — QFT · tensors-symmetric-antisymmetric

Problem

A second-rank tensor is called *symmetric* if it is unchanged when you switch the indices (

s^{\nu\mu} = s^{\mu\nu}

); it is called *antisymmetric* if it changes sign (

a^{\nu\mu} = -a^{\mu\nu}

). (a) How many independent elements are there in a symmetric tensor? (Since

s^{12} = s^{21}

, these would count as only *one* independent element.) (b) How many independent elements are there in an antisymmetric tensor? (c) If

s^{\mu\nu}

is symmetric, show that

s_{\mu\nu}

is also symmetric. If

a^{\mu\nu}

is antisymmetric, show that

a_{\mu\nu}

is antisymmetric. (d) If

s^{\mu\nu}

is symmetric and

a^{\mu\nu}

is antisymmetric, show that

s_{\mu\nu}a^{\mu\nu} = 0

. (e) Show that any second-rank tensor (

t^{\mu\nu}

) can be written as the sum of an antisymmetric part (

a^{\mu\nu}

) and a symmetric part (

s^{\mu\nu}

t^{\mu\nu} = a^{\mu\nu} + s^{\mu\nu}

. Construct

s^{\mu\nu}

and

a^{\mu\nu}

explicitly, given

t^{\mu\nu}