Delta written as Minkowski metric ?

[binbagsss]

Homework Statement

Hi, I am just stuck in why / how we can write minkowski metric where I would usually write delta.

I see that the product rule is used in the first term to cancel the terms in the second term since partials commute for a scalar and so we are left with the d rivative acting on a x terms.

I would usually write $\frac{\partial x^v}{\partial x ^y} = \delta ^{(4)v} _y$

How is this equivalent to the minkowski metric ?

Homework Equations

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The Attempt at a Solution

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Many thanks [/B]

 

[stevendaryl]

Hi, I am just stuck in why / how we can write minkowski metric where I would usually write delta.

The metric comes into play because:

$[p_\mu, x^\nu] = -i \hbar \delta^\nu_\mu$ as you would expect, but

$[p_\mu, x_\nu] = -i \hbar \eta_{\nu \mu}$

The $\eta_{\nu \mu}$ is part of the definition of $x_\nu$: $x_\nu = \eta_{\nu \lambda} x^\lambda$