Re: Lagrange multiplier in Plebanski action

In article <slrnesli6e.pi.igor.kh@xxxxxxxxxxxxxxxxxxx>,
Igor Khavkine <igor.kh@xxxxxxxxx> wrote:

On 2007-02-08, yyoon@xxxxxxxxxxxxxxx <yyoon@xxxxxxxxxxxxxxx> wrote:

What is this Lagrange multiplier?
I know Lagrange multiplier from my multivariable calculus class, but
do not really understand what this Lagrange multiplier is in this
Lagrangian action.

It's the same thing.

Igor has given a perfectly fine answer, but maybe something more
elementary might also help. A "Lagrange multiplier" is a trick
for minimizing or maximizing something *while making sure some
constraint holds*. The idea is that if you add a term like

lambda x stuff

to the quantity being extremized, and you take the derivative with
respect to lambda and set this equal to zero, you get the constraint
equation

stuff = 0

as long as the quantity lambda appears nowhere else in the quantity
being extremized.

In field theory, we call lambda a "Lagrange multiplier field". We call
lambda x stuff a "Lagrange multiplier term".

It's presumably no coincidence that "Lagrange multiplier fields" show up
in "Lagrangians".

Is this something like gauge fixing term?

Gauge fixing terms are an example of this trick, where the constraint
equation you're trying to impose is of the "gauge fixing" sort.

All this stuff shows up in path integral quantization, too. But, it's
trickier. Some relevant buzzwords are "Fadeev-Popov", "ghosts", and
"BRST".


.

Relevant Pages

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