Proof involving exponential of anticommuting operators

• Joker93
In summary, the speaker has been able to prove a specific case for a given value of N, but is struggling to show the result for N>1. They mention a generalization of their initial proof for N=1, and how they used this to substitute into the final result. However, they found it difficult to reach a meaningful result for the N>1 case. They also mention using a formula from Fradkin's book, but are unsure if this is where they went wrong. They ask for any help or guidance on the matter.
Joker93
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.

Note that I have proved that the last relation holds.
If anybody could help with the problem by just even a suggestion, It would be appreciated.

Last edited:

1. What is an anticommuting operator?

An anticommuting operator is a type of operator in mathematics and physics that satisfies the property of anticommutation, which means that the order in which two operators are multiplied matters. In other words, if A and B are anticommuting operators, then AB = -BA.

2. How is the exponential of an anticommuting operator defined?

The exponential of an anticommuting operator is defined using the Taylor series expansion, which is a mathematical representation of a function as an infinite sum of terms. The exponential of an anticommuting operator is given by the infinite sum of all the powers of the operator, with alternating signs.

3. What is the significance of studying proofs involving exponential of anticommuting operators?

Proofs involving exponential of anticommuting operators are important in the fields of quantum mechanics and statistical mechanics, as they are used to describe the behavior of particles and systems at the microscopic level. They also have applications in other areas of physics and mathematics, such as in the study of Lie algebras and group theory.

4. Can you give an example of a proof involving exponential of anticommuting operators?

One example of a proof involving exponential of anticommuting operators is the proof of the Baker-Campbell-Hausdorff formula, which is used to simplify the multiplication of two exponential operators. This formula is frequently used in quantum mechanics to calculate the time evolution of a system.

5. Are there any applications of proofs involving exponential of anticommuting operators in other fields?

Yes, proofs involving exponential of anticommuting operators have applications in various fields such as computer science, where they are used in the development of quantum algorithms and quantum computing. They are also used in signal processing, control theory, and other areas of mathematics and engineering.

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