Complex Scalar Field in Terms of Two Independent Real Fields

[ghotra]

I am working with a complex scalar field written in terms of two independent real scalar fields and trying to derive the commutator relations.

So,

$$\phi = \frac{1}{\sqrt{2}} \left(\phi_1 + i \phi_2)$$

where $\phi_1$ and $\phi_2$ are real.

When deriving,

$$[\phi(\vec{x},t),\dot{\phi}(\vec{x}',t)] = 0$$

I get terms like the following:

$$[\phi_1(\vec{x},t),\dot{\phi}_2(\vec{x}',t)]$$

which I need to vanish. It makes sense to me that they should vanish, but how do I show this?

 

[ghotra]

Hmm...I think that we just take that as the quantization condition. That is,

$$[\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t}] = i \delta^3(\vec{x}-\vec{x}{\,}')\delta_{rs}$$

Is this correct?

 

[SpaceTiger]

Hmm...I think that we just take that as the quantization condition. That is,

$$[\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t}] = i \delta^3(\vec{x}-\vec{x}{\,}')\delta_{rs}$$

Is this correct?

Since $\phi_1$ and $\phi_2$ are independent, they'll only be canonically conjugate with their own momenta (the $\delta_{rs}$ on the left). Your equation just states that in combination with the usual commutation relation of the real scalar field.

 

[snooper007]

$$\phi_1$$ and $$\phi_2$$
are independent fields, so
$$[\phi_1, \dot{\phi}_2]$$=0

 

[dextercioby]

What is the Poisson bracket between the classical fields ? If you know that, you can canonically quantize using Dirac's rule.

Daniel.