Complex Scalar Field in Terms of Two Independent Real Fields


by ghotra
Tags: complex, field, fields, independent, real, scalar, terms
ghotra
ghotra is offline
#1
Oct22-05, 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?
Phys.Org News Partner Physics news on Phys.org
Physicists design quantum switches which can be activated by single photons
'Dressed' laser aimed at clouds may be key to inducing rain, lightning
Higher-order nonlinear optical processes observed using the SACLA X-ray free-electron laser
ghotra
ghotra is offline
#2
Oct23-05, 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?
SpaceTiger
SpaceTiger is offline
#3
Oct23-05, 10:15 AM
Emeritus
Sci Advisor
PF Gold
SpaceTiger's Avatar
P: 2,977
Quote Quote by ghotra
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?
Since [itex]\phi_1[/itex] and [itex]\phi_2[/itex] are independent, they'll only be canonically conjugate with their own momenta (the [itex]\delta_{rs}[/itex] on the left). Your equation just states that in combination with the usual commutation relation of the real scalar field.

snooper007
snooper007 is offline
#4
Oct25-05, 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
dextercioby
dextercioby is offline
#5
Oct26-05, 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.


Register to reply

Related Discussions
DEC with E/M and scalar fields Special & General Relativity 4
Do you think that scalar electromagnetic weapons are real and possible? General Discussion 1
Connection between Scalar fields and Field Thoeries Special & General Relativity 0
Is the usual non-abelian gauge field A real or complex? Quantum Physics 1
Is Yong-Mills field real or complex? Beyond the Standard Model 0