# Symmetry of the Green-Keldysh-Nambu function

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• Paul159
In summary: Your Name]In summary, the forum user is asking for clarification on a symmetry stated in Volkov's article regarding the Green function for spin-up and spin-down particles. The symmetry is related to the time-reversal symmetry of the system, where the Green function is invariant under the transformation ##\bar{1} = 2, \bar{2} = 1##. This is due to the absence of external fields and spin-orbit interactions. The user's example is not quite correct and the correct equation should be $$\langle \Psi_\uparrow (1_1)^\dagger \Psi_\uparrow(2_1) \rangle = \langle \Psi_\downarrow (1_2)^\ Paul159 Hello, I would like to understand a relation of this article by Volkov (eq. 4). Let's define the Green function$$ G^{ij}_{ab} (1,2) = -i \langle T_c \Psi_a (1_i) \Psi_b (2_j) \rangle $$where ##a,b = (1,2)## are the spin indices and ##i,j = (1,2) ## are the indices for the Keldysh contour ; ##\Psi_1 = \Psi_\uparrow##, ##\Psi_2 = \Psi_\downarrow^\dagger##. ##T_c## is the contour ordering. Now I cite Volkov : "if the system contains no fields acting directly on the spins and if the spin-orbit interaction can be neglected we have :$$ G^{ij*}_{ab} (1,2) = G^{\bar{ij}}_{\bar{ab}} (1,2) $$where ##\bar{1} = 2, \bar{2} = 1##. " I don't understand why this last symmetry is true. For example I get with the last equation that$$ \langle \Psi_\uparrow (1_1)^\dagger \Psi_\uparrow(2_1) \rangle = \langle \Psi_\downarrow (1_2)^\dagger \Psi_\downarrow(2_2) \rangle ,$$where 1 is after 2 on the first branch. Thanks in advance for any help. Thank you for your question about the relation in Volkov's article. It appears that the symmetry you are referring to is related to the time-reversal symmetry of the system. In the absence of external fields and spin-orbit interactions, the time-reversal symmetry of the system is preserved. This means that the Green function should be invariant under the transformation ##\bar{1} = 2, \bar{2} = 1##, as stated in the equation you cited. To understand this symmetry, let's consider the time evolution of the system. Since there are no external fields acting directly on the spins, the time evolution of the spin operators will be the same on both branches of the Keldysh contour. This means that the Green function for spin-up and spin-down particles should be the same on both branches, leading to the symmetry stated in Volkov's article. To address your example, the equation you wrote is not quite correct. The correct equation should be$$ \langle \Psi_\uparrow (1_1)^\dagger \Psi_\uparrow(2_1) \rangle = \langle \Psi_\downarrow (1_2)^\dagger \Psi_\downarrow(2_1) \rangle , where the spin operators on the first branch are the same for both spin-up and spin-down particles. This is consistent with the time-reversal symmetry of the system.

I hope this explanation helps you understand the relation in Volkov's article. If you have any further questions, please don't hesitate to ask. Thank you for your interest in this topic.

## 1. What is the Green-Keldysh-Nambu function?

The Green-Keldysh-Nambu function is a mathematical function used in condensed matter physics to describe the dynamics of quantum systems. It combines the Green's function, which describes the propagation of particles in a system, with the Keldysh contour, which allows for the description of non-equilibrium systems. The Nambu space, named after Japanese physicist Yoichiro Nambu, is a mathematical framework used to describe systems with both particle-like and hole-like excitations.

## 2. Why is the symmetry of the Green-Keldysh-Nambu function important?

The symmetry of the Green-Keldysh-Nambu function is important because it is a fundamental property that dictates the behavior of the function and the system it describes. Symmetry can reveal important information about the system, such as the presence of certain conservation laws or underlying physical principles.

## 3. What are the different types of symmetry in the Green-Keldysh-Nambu function?

There are three types of symmetry in the Green-Keldysh-Nambu function: time-reversal symmetry, charge-conjugation symmetry, and particle-hole symmetry. Time-reversal symmetry states that the function remains unchanged when time is reversed. Charge-conjugation symmetry states that the function remains unchanged when particles are replaced with their antiparticles. Particle-hole symmetry states that the function remains unchanged when particles and holes are interchanged.

## 4. How is the symmetry of the Green-Keldysh-Nambu function related to physical observables?

The symmetry of the Green-Keldysh-Nambu function is related to physical observables through Noether's theorem, which states that for every continuous symmetry in a physical system, there is a corresponding conserved quantity. This means that the symmetry of the function can provide information about the behavior of physical observables in the system.

## 5. How is the symmetry of the Green-Keldysh-Nambu function studied and applied in research?

The symmetry of the Green-Keldysh-Nambu function is studied and applied in research through various theoretical and computational methods. These include group theory, which is used to analyze the symmetry of the function, and numerical simulations, which can provide insights into the behavior of physical systems described by the function. Understanding the symmetry of the function can also aid in the development of new materials and technologies in the field of condensed matter physics.

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