- #1
luksen
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Hi,
for my exam i"m re-reading Peskin&Schroeder and stumbled across equations 2.21-2.25 where the canonical quantization of the KG field is done.
P&S start with doing a Fourier trf on [tex]\phi(x,t)=\int\frac{d^3p}{(2\pi)^3}e^{ip\cdot x}\phi(p,t)[/tex]
applying the KG operator in that results in [tex](\frac{\partial^2}{\partial t^2}+|p|+m^2)\phi(p,t)=0[/tex]
P&S go on to recollect hte SHO where [tex]\phi=\frac{1}{\sqrt{2\omega}}(a+a^\dagger)[/tex]
so P&S say that in analogy you arrive at
[tex]\phi(x,t)=\int\frac{d^3p}{(2\pi)^3}(a_pe^{ip\cdot x}+a_p^\dagger e^{-ip\cdot x})[/tex]
but straight forward subsitution would yield with no negative frequency terms
[tex]\phi(x,t)=\int\frac{d^3p}{(2\pi)^3}(a_pe^{ip\cdot x}+a_p^\dagger e^{+ip\cdot x})[/tex]
I've seen this expansion derived differently and understand it when i follo w it but using this SHO analogy i can't follow thie last step to 2.25
i'd be grateful for input
for my exam i"m re-reading Peskin&Schroeder and stumbled across equations 2.21-2.25 where the canonical quantization of the KG field is done.
P&S start with doing a Fourier trf on [tex]\phi(x,t)=\int\frac{d^3p}{(2\pi)^3}e^{ip\cdot x}\phi(p,t)[/tex]
applying the KG operator in that results in [tex](\frac{\partial^2}{\partial t^2}+|p|+m^2)\phi(p,t)=0[/tex]
P&S go on to recollect hte SHO where [tex]\phi=\frac{1}{\sqrt{2\omega}}(a+a^\dagger)[/tex]
so P&S say that in analogy you arrive at
[tex]\phi(x,t)=\int\frac{d^3p}{(2\pi)^3}(a_pe^{ip\cdot x}+a_p^\dagger e^{-ip\cdot x})[/tex]
but straight forward subsitution would yield with no negative frequency terms
[tex]\phi(x,t)=\int\frac{d^3p}{(2\pi)^3}(a_pe^{ip\cdot x}+a_p^\dagger e^{+ip\cdot x})[/tex]
I've seen this expansion derived differently and understand it when i follo w it but using this SHO analogy i can't follow thie last step to 2.25
i'd be grateful for input