Thermal states and black holes I

-

Thermal equilibrium state 

$<\bar{X}>_T = \frac{1}{Z} Tr(\bar{X} e^{- \beta H})$

                         =$Tr(\bar{X \rho_T})$

where $\rho_T = \rho_{\beta} = \frac{e^{- \beta H}}{Z}$

  and         $Z = \sum_i e^{- \beta E_i} $


Double Language, Entanglement, and Umezawa -- Thermofield Double

$\psi = \frac{1}{Z} \sum_n e^{- \frac{\beta}{2} E_n} |n\rangle_1 \otimes |n \rangle_2$

For 

$\bar{X}_1$

$\langle \psi | \bar{X}_1 | \psi \rangle = \frac{1}{Z} \sum_n  e^{- \beta E_n} \langle n|\bar{X}| n \rangle  = <\bar{X}>$

$Tr_2(| \psi \rangle \langle \psi|) = \frac{1}{Z} \sum_n  e^{- \beta E_n}|n\rangle_1 {_1}\langle n|$

                                                   $= \rho_\beta$

Comments

Post a Comment

Popular posts from this blog

Dark Matter and Particle Mass Bounds --- Quick notes II

-
Lower bound on bosonic dark matter                    - Use the uncertainty principle                                             pack all particles into one cell of volume                            -$\delta x \delta p$~ $ 1$                                   -$\ 2R_{halo} \times m \times v$~ $ 1$                      $m_{boson} \geq 10^{-23}$  given some data Lower bound on Fermionic dark matter                          - From fermi exclusion spread out particles throughout the volume                                   $M_{halo} =m_{fermion} V \int f(p) d^3 p$                      $f(p) = 1$              -Add data          $m_{fermion} \geq 10eV$                                
Read more

INTRODUCTION

-
I could not be more pleased to announce that this space of the internet will be dedicated to whatever physics topics I stumble upon in the day and get motivated enough to try to understand. I hope to engage in conversation with plenty of people over these topics. There are no rules here and there is no judgment of any form. Just type away any of your observations and comments on the content.
Read more

Quick notes on Supersymmetric Quantum Mechanics I

-
Bosonic Harmonic Oscillator: $H = a a^{\dagger} + \frac{1}{2}$ Where   $a = \frac{1}{\sqrt{2}}(p - i x)$ $a^{\dagger} = \frac{1}{\sqrt{2}}(p - i x)$ with commutation relations : $[a^{\dagger}, a] =1$ $[a^{\dagger}, a^{\dagger}] =0$ $[a, a] =0$ Fermionic Harmonic Oscillator: $H = b b^{\dagger} - \frac{1}{2}$ with anticommutation relations $\{b^{\dagger}, b^{\dagger}\} =0$ $\{b, b\}=0$ Consider:  $H = aa^{\dagger} + bb^{\dagger}$ with  Fock space:   $| \#bosons, \#fermions \rangle$ One seeks for example: $Q|1, 0 \rangle = |1, 1 \rangle$ $Q = a b^{\dagger}$ $Q^{\dagger} = b a^{\dagger}$ $Q$ is called a supercharge. Bosonic state: Even number of fermions Fermionic state: Odd number of fermions Consider $N = 2 $ supersymmetry $Q_1|boson 1 \rangle = |Fermion 1 \rangle$ $Q_2|boson 1 \rangle = |Fermion 2 \rangle$ $Q_1^{\dagger}|Fermion 1\rangle = |boson 1 \rangle$ $Q_2^{\dagger}|Fermion 1 \rangle = |boson 2\rangle$ Supersymmetry can be broken      - Spontaneously broken             - $[H, Q] =
Read more