Grassman number

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Main Question or Discussion Point

If you have a Grassman number [tex]\eta[/tex] that anticommutes with the creation and annihilation operators, then is the expression:

[tex]<0|\eta|0>[/tex]

well defined? Because you can write this as:

[tex]<1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1>
=-<1|\eta|1>[/tex]

But if [tex]\eta[/tex] is a constant, then shouldn't:


[tex]<0|\eta|0>=<1|\eta|1>=\eta[/tex] ?
 

Answers and Replies

  • #2
A. Neumaier
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If you have a Grassman number [tex]\eta[/tex] that anticommutes with the creation and annihilation operators, then is the expression:

[tex]<0|\eta|0>[/tex]

well defined? Because you can write this as:

[tex]<1|a^{\dagger} \eta a|1>=-<1| \eta a^{\dagger} a|1>
=-<1|\eta|1>[/tex]

But if [tex]\eta[/tex] is a constant, then shouldn't:


[tex]<0|\eta|0>=<1|\eta|1>=\eta[/tex] ?
Grassmann numbers are operators (though they are called numbers).
[tex]<0|\eta|0>=0[/tex] is well-defined and vanishes.
 

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