- #1
physichu
- 30
- 1
Puting a minus in front of the momentum in the field expansion gives
##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{ - {\bf{p}}}}{e^{ - i{\bf{p}} \cdot {\bf{x}}}} + a_{ - {\bf{p}}}^ + {e^{i{\bf{p}} \cdot {\bf{x}}}}} \right)##.
Is this implise that
##a_{ - {\bf{p}}}^ + = {a_{{\bf{p}}{\rm{ }}}}## ##{a_{ - {\bf{p}}}} = a_{\bf{p}}^ + ## ?
Becuse if so
##\displaylines{
\pi \left( {\bf{x}} \right) = - i\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}\sqrt {{{{\omega _{\bf{p}}}} \over 2}} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} - a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right)} = \cr
= - i\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}\sqrt {{{{\omega _{\bf{p}}}} \over 2}} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} - a_{ - {\bf{p}}}^{}{e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right) = } \cr
= - i\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}\sqrt {{{{\omega _{\bf{p}}}} \over 2}} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} - a_{\bf{p}}^{}{e^{i{\bf{p}} \cdot {\bf{x}}}}} \right)} = 0 \cr} ##
Wich is obviosly wrong, Where is the mistake?
##\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} + a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right){\rm{ }}\phi \left( {\bf{x}} \right) = \int {{d^3}\tilde p} \left( {{a_{ - {\bf{p}}}}{e^{ - i{\bf{p}} \cdot {\bf{x}}}} + a_{ - {\bf{p}}}^ + {e^{i{\bf{p}} \cdot {\bf{x}}}}} \right)##.
Is this implise that
##a_{ - {\bf{p}}}^ + = {a_{{\bf{p}}{\rm{ }}}}## ##{a_{ - {\bf{p}}}} = a_{\bf{p}}^ + ## ?
Becuse if so
##\displaylines{
\pi \left( {\bf{x}} \right) = - i\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}\sqrt {{{{\omega _{\bf{p}}}} \over 2}} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} - a_{\bf{p}}^ + {e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right)} = \cr
= - i\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}\sqrt {{{{\omega _{\bf{p}}}} \over 2}} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} - a_{ - {\bf{p}}}^{}{e^{ - i{\bf{p}} \cdot {\bf{x}}}}} \right) = } \cr
= - i\int {{{{d^3}p} \over {{{\left( {2\pi } \right)}^3}}}\sqrt {{{{\omega _{\bf{p}}}} \over 2}} \left( {{a_{\bf{p}}}{e^{i{\bf{p}} \cdot {\bf{x}}}} - a_{\bf{p}}^{}{e^{i{\bf{p}} \cdot {\bf{x}}}}} \right)} = 0 \cr} ##
Wich is obviosly wrong, Where is the mistake?