- #1
Joker93
- 502
- 37
- Homework Statement
- The problem is to prove equation (5.31) from the book "Quarks, gluons and lattices" by Creutz. It involves anticommuting operators and functions of them acting on defined states, which I give below.
- Relevant Equations
- On page 23 of the book "Quarks, gluons and lattices" by Creutz, he defines a state
$$\langle\psi|=\langle 0|e^{bFc}e^{\lambda b^\dagger G c^\dagger}$$
where ##\lambda## is a number, ##F, G## are ##N\times N## symmetric matrices and ##b, c## are vectors whose components ##b_m, c_m## are operators such that their anticommutators satisfy
$$\{b_m^\dagger, b_n\}=\{c_m^\dagger, c_n\}=\delta_{mn}$$
with every other anti-commutator being zero, and the state ##\langle 0 |## such that
$$\langle 0 |c^\dagger_m=\langle 0 |b^\dagger_m=0$$
Creutz says that a straightfoward calculation can lead us to proving that
$$\langle\psi|b^\dagger=-\langle\psi|(F^{-1}-\lambda G)^{-1} c=-\langle\psi|(1-\lambda FG)^{-1}Fc$$
For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that
$$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$
which generalizes what I initially found for the ##N=1## case, which is ##\langle\psi|b^\dagger=-\langle 0|Fc##. For this last result for the ##N=1## case, I then substituted ##\langle 0 |=\langle \psi | e^{-\lambda b^\dagger G c^\dagger}e^{-bFc}##, and after some manipulations, I reached the end of the proof. Doing something similar for the ##N>1## case, I found it too difficult to get to a meaningful result (or even close to the final result).
Note that for the ##N>1## case, I have used that, for example,
$$e^{bFc}=e^{\sum_{ij}b_i F_{ij} c_j}=\prod_{ij}(1+b_iF_{ij}c_j)$$
which is found on page 193 of Fradkin's book "Quantum Field theory: an integrated approach". Note that there, Fradkin was talking about Grassmann variables being on the exponential, whereas here we have the above non-trivial anticommutation relations; so, this might have been where I got it wrong.
If anybody can give a hint or some guidance on this, or even provide with some reference that can help, it would be greatly appreciated.
$$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$
which generalizes what I initially found for the ##N=1## case, which is ##\langle\psi|b^\dagger=-\langle 0|Fc##. For this last result for the ##N=1## case, I then substituted ##\langle 0 |=\langle \psi | e^{-\lambda b^\dagger G c^\dagger}e^{-bFc}##, and after some manipulations, I reached the end of the proof. Doing something similar for the ##N>1## case, I found it too difficult to get to a meaningful result (or even close to the final result).
Note that for the ##N>1## case, I have used that, for example,
$$e^{bFc}=e^{\sum_{ij}b_i F_{ij} c_j}=\prod_{ij}(1+b_iF_{ij}c_j)$$
which is found on page 193 of Fradkin's book "Quantum Field theory: an integrated approach". Note that there, Fradkin was talking about Grassmann variables being on the exponential, whereas here we have the above non-trivial anticommutation relations; so, this might have been where I got it wrong.
If anybody can give a hint or some guidance on this, or even provide with some reference that can help, it would be greatly appreciated.