Do All Gauge Fields Intrinsically Contribute to the Lagrangian?

[Living_Dog]

After deriving the Lagrangian for the electromagnetic field using only gauge invariance of the action, the result is: (i.e. to say $$\delta L \equiv 0$$.)

$$L \equiv (\partial_{\mu} + ieA_{\mu})\phi(\partial^{\mu} - ieA^{\mu})\phi^* - m^2\phi^2$$.

Et. viola'. Done. Finished. Complete. Let's go home and party till it's 1999.

WRONG CAMEL BREATH!

Ryder then pulls out a statement from some orifice unbeknown to ex-graduate wanna-be physicist students, to wit:

"The field $$A_{\mu}$$, however, must presumably contribute by itself to the Lagrangian."
​

When I read that I heard a very loud "pop" as if the entrance to said orifice was suddenly opened and closed. So a big hug to the person who defends... I mean explains this in Ryder!:)

Ok, all joking aside, I know we need the electromagnetic field strength tensor in order to recover E&M from the Lagrangian when we vary $$A_{\mu}$$. But the above statement is so ad hoc and short that if this was some unknown field no one would know to add any other term whatsoever. ... no?

-ld

 

[martinbn]

Isn't that because it has it interact with itself?

 

[Living_Dog]

Isn't that because it has it interact with itself?

That term is necessary to include the E- and B-fields in the quantized version of E&M.

Here is a thread which discusses this term: https://www.physicsforums.com/archive/index.php/t-37318.html

The terms which contain the fields themselves, such as $$\phi$$, $$A_\mu$$ and $$\phi^*$$ are interaction terms. I think the derivative terms are about how the fields change in spacetime - the dynamics of the fields.

 

[arkajad]

You should start with asking: what is my system that I want to describe? Is just the complex field $\phi$? Or is it a composite system consisting of

a) e-m field
b) interacting with a another field, $\phi$

If you want only b) - yes, go home. But Ryder doesn't want to go home without dealing also with a). Some other author will still stay before leaving, as he/she may want to add

c) interacting wit a) an b) also ever-present gravitational field.

Will everybody be happy now? Not at all. What about "vacuum fluctuations?" And so on.

 

[Living_Dog]

You should start with asking: what is my system that I want to describe? Is just the complex field $\phi$? Or is it a composite system consisting of

a) e-m field
b) interacting with a another field, $\phi$

If you want only b) - yes, go home. But Ryder doesn't want to go home without dealing also with a). Some other author will still stay before leaving, as he/she may want to add

c) interacting wit a) an b) also ever-present gravitational field.

Will everybody be happy now? Not at all. What about "vacuum fluctuations?" And so on.

[1] The F_\mu\nu term does not interact with \phi. He added it b/c A_\mu "...contributes by itself..." to the Lagrangian.

[2] What is the motivation to say a gauge field "...contributes by itself..." to the Lagrangian? Do all gauge fields contribute by themselves to the Lagrangian? Are there similar terms in the electro-weak Lagrangian?? (ooops, I may have answered my own question if the answer is 'yes'!)

[3] I still don't get "vacuum fluctuations" - to me they are a cult religion borne out of our lack of understanding that nothing is nothing and the confusion over the zero-point energy (thank you very much harmonic oscillator!). a+|0> = |1> ! Oh, isn't this Genesis 1:1?? :)


Thanks for the replies!

 

[arkajad]

Do all gauge fields contribute by themselves to the Lagrangian?

Somehow you managed not to notice the essence. Which Lagrangian? Of which system? If the author is not quite clear about what he has in mind, you are supposed to turn on your own thinking. Otherwise readings textbooks would be a boring activity. And it does not have to be such.