
#1
Oct2205, 10:33 PM

P: 53

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, [tex] \phi = \frac{1}{\sqrt{2}} \left(\phi_1 + i \phi_2) [/tex] where [itex]\phi_1[/itex] and [itex]\phi_2[/itex] are real. When deriving, [tex] [\phi(\vec{x},t),\dot{\phi}(\vec{x}',t)] = 0 [/tex] I get terms like the following: [tex][\phi_1(\vec{x},t),\dot{\phi}_2(\vec{x}',t)][/tex] which I need to vanish. It makes sense to me that they should vanish, but how do I show this? 



#2
Oct2305, 01:03 AM

P: 53

Hmm...I think that we just take that as the quantization condition. That is,
[tex] [\phi_r(\vec{x},t),\pi_s(\vec{x}{\,}',t}] = i \delta^3(\vec{x}\vec{x}{\,}')\delta_{rs} [/tex] Is this correct? 



#3
Oct2305, 10:15 AM

Emeritus
Sci Advisor
PF Gold
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#4
Oct2505, 08:51 PM

P: 36

Complex Scalar Field in Terms of Two Independent Real Fields
[tex]\phi_1[/tex] and [tex]\phi_2[/tex]
are independent fields, so [tex][\phi_1, \dot{\phi}_2][/tex]=0 



#5
Oct2605, 05:33 AM

Sci Advisor
HW Helper
P: 11,863

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


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