Advanced Quantum Mechanics II PHYS 40202

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To be used in future

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A.6 Relativity-tensor notation

For relativistic calculation we use the Bjorken-Drell metric↓↓

and the four-vector notation↓

Lowered indices:↓

where we use the full Einstein summation convention↓ that “any index that appears as upper and lower index is summed over; indices can be raised and lowered by the metric tensor↓”. Thus

and

In the same way

(

is the Dalembertian↓).

We also find that

where the “Kronecker

”↓ is one if both indices are equal, and zero otherwise.

We shall denote a Lorentz transformation↓ by

, and as we change inertial frames↓ a four vector transforms as

where the position of the indices on

is crucial. From

and the invariance of

we find

(Note the change of place of the indices). What it means is that the only difference is a minus sign for the components in the

and

entries of the matrix.

Prev Section A.5: Matrix algebra Up Appendix A: Useful Mathematics Section A.7: Commutators Next