Gell-Mann and Low theorem

In quantum field theory, the Gell-Mann and Low theorem is a mathematical statement that allows one to relate the ground (or vacuum) state of an interacting system to the ground state of the corresponding non-interacting theory. It was proved in 1951 by Murray Gell-Mann and Francis E. Low. The theorem is useful because, among other things, by relating the ground state of the interacting theory to its non-interacting ground state, it allows one to express Green's functions (which are defined as expectation values of Heisenberg-picture fields in the interacting vacuum) as expectation values of interaction picture fields in the non-interacting vacuum. While typically applied to the ground state, the Gell-Mann and Low theorem applies to any eigenstate of the Hamiltonian. Its proof relies on the concept of starting with a non-interacting Hamiltonian and adiabatically switching on the interactions.

History

The theorem was proved first by Gell-Mann and Low in 1951, making use of the Dyson series. ^[1] In 1969, Klaus Hepp provided an alternative derivation for the case where the original Hamiltonian describes free particles and the interaction is norm bounded. ^[2] In 1989, G. Nenciu and G. Rasche proved it using the adiabatic theorem. ^[3] A proof that does not rely on the Dyson expansion was given in 2007 by Luca Guido Molinari. ^[4]

Statement of the theorem

Let ${\displaystyle |\Psi _{0}\rangle }$ be an eigenstate of ${\displaystyle H_{0}}$ with energy ${\displaystyle E_{0}}$ and let the 'interacting' Hamiltonian be ${\displaystyle H=H_{0}+gV}$, where ${\displaystyle g}$ is a coupling constant and ${\displaystyle V}$ the interaction term. We define a Hamiltonian ${\displaystyle H_{\epsilon }=H_{0}+e^{-\epsilon |t|}gV}$ which effectively interpolates between ${\displaystyle H}$ and ${\displaystyle H_{0}}$ in the limit ${\displaystyle \epsilon \rightarrow 0^{+}}$ and ${\displaystyle |t|\rightarrow \infty }$. Let ${\displaystyle U_{\epsilon I}}$ denote the evolution operator in the interaction picture. The Gell-Mann and Low theorem asserts that if the limit as ${\displaystyle \epsilon \rightarrow 0^{+}}$ of

${\displaystyle |\Psi _{\epsilon }^{(\pm )}\rangle ={\frac {U_{\epsilon I}(0,\pm \infty )|\Psi _{0}\rangle }{\langle \Psi _{0}|U_{\epsilon I}(0,\pm \infty )|\Psi _{0}\rangle }}}$

exists, then ${\displaystyle |\Psi _{\epsilon }^{(\pm )}\rangle }$ are eigenstates of ${\displaystyle H}$.

When applied to, say, the ground-state, the theorem does not guarantee that the evolved state will be a ground state. In other words, level crossing is not excluded.

Proof

As in the original paper, the theorem is typically proved making use of Dyson's expansion of the evolution operator. Its validity however extends beyond the scope of perturbation theory as has been demonstrated by Molinari. We follow Molinari's method here. Focus on ${\displaystyle H_{\epsilon }}$ and let ${\displaystyle g=e^{\epsilon \theta }}$. From Schrödinger's equation for the time-evolution operator

${\displaystyle i\hbar \partial _{t_{1}}U_{\epsilon }(t_{1},t_{2})=H_{\epsilon }(t_{1})U_{\epsilon }(t_{1},t_{2})}$

and the boundary condition ${\displaystyle U_{\epsilon }(t_{2},t_{2})=1}$ we can formally write

${\displaystyle U_{\epsilon }(t_{1},t_{2})=1+{\frac {1}{i\hbar }}\int _{t_{2}}^{t_{1}}dt'(H_{0}+e^{\epsilon (\theta -|t'|)}V)U_{\epsilon }(t',t_{2}).}$

Focus for the moment on the case ${\displaystyle 0\geq t_{1}\geq t_{2}}$. Through a change of variables ${\displaystyle \tau =t'+\theta }$ we can write

${\displaystyle U_{\epsilon }(t_{1},t_{2})=1+{\frac {1}{i\hbar }}\int _{\theta +t_{2}}^{\theta +t_{1}}d\tau (H_{0}+e^{\epsilon \tau }V)U_{\epsilon }(\tau -\theta ,t_{2}).}$

We now demonstrate that ${\displaystyle \epsilon g\partial _{g}U_{\epsilon }(t_{1},t_{2})=\partial _{t_{1}}U_{\epsilon }(t_{1},t_{2})+\partial _{t_{2}}U_{\epsilon }(t_{1},t_{2})}$.

Consider a g-independent operator ${\displaystyle H'(t)=H_{0}+e^{\epsilon t}V}$ and its corresponding evolution operator ${\displaystyle U'(t_{1},t_{2})}$. We can write

${\displaystyle U_{\epsilon }(t_{1},t_{2})=1+{\frac {1}{i\hbar }}\int _{\theta +t_{2}}^{\theta +t_{1}}d\tau H'(\tau )U_{\epsilon }(\tau -\theta ,t_{2}).}$

Using the same technique with ${\displaystyle U_{\epsilon }(t_{1},t_{2})}$, write the equation for ${\displaystyle U'(t_{1}+\theta ,t_{2}+\theta )}$ as

${\displaystyle U'(t_{1}+\theta ,t_{2}+\theta )=1+{\frac {1}{i\hbar }}\int _{\theta +t_{2}}^{\theta +t_{1}}d\tau H'(\tau )U_{\epsilon }(\tau ,t_{2}+\theta ).}$

The uniqueness of solution implies the identification

${\displaystyle U_{\epsilon }(t_{1},t_{2})=U'(t_{1}+\theta ,t_{2}+\theta ).}$

Combining thees results, we obtain

${\displaystyle \partial _{\theta }U_{\epsilon }(t_{1},t_{2})=\epsilon g\partial _{g}U_{\epsilon }(t_{1},t_{2})=\partial _{t_{1}}U_{\epsilon }(t_{1},t_{2})+\partial _{t_{2}}U_{\epsilon }(t_{1},t_{2}).}$

This result can be combined with the Schrödinger equation and its adjoint

${\displaystyle -i\hbar \partial _{t_{1}}U_{\epsilon }(t_{2},t_{1})=U_{\epsilon }(t_{2},t_{1})H_{\epsilon }(t_{1})}$

to obtain

${\displaystyle i\hbar \epsilon g\partial _{g}U_{\epsilon }(t_{1},t_{2})=H_{\epsilon }(t_{1})U_{\epsilon }(t_{1},t_{2})-U_{\epsilon }(t_{1},t_{2})H_{\epsilon }(t_{2}).}$

The corresponding equation between ${\displaystyle H_{\epsilon I},U_{\epsilon I}}$ is the same. It can be obtained by pre-multiplying both sides with ${\displaystyle e^{iH_{0}t_{1}/\hbar }}$, post-multiplying with ${\displaystyle e^{iH_{0}t_{2}/\hbar }}$ and making use of

${\displaystyle U_{\epsilon I}(t_{1},t_{2})=e^{iH_{0}t_{1}/\hbar }U_{\epsilon }(t_{1},t_{2})e^{-iH_{0}t_{2}/\hbar }.}$

${\displaystyle H_{\epsilon I}(t)=e^{iH_{0}t/\hbar }H_{\epsilon }(t)e^{-iH_{0}t/\hbar }}$

The other case we are interested in, namely ${\displaystyle t_{2}\geq t_{1}\geq 0}$ can be treated in an analogous fashion and yields an additional minus sign in front of the commutator (we are not concerned here with the case where ${\displaystyle t_{1,2}}$ have mixed signs). In summary, we obtain

${\displaystyle \left(H_{\epsilon I,t=0}-E_{0}\pm i\hbar \epsilon g\partial _{g}\right)U_{\epsilon I}(0,\pm \infty )|\Psi _{0}\rangle =0.}$

We proceed for the negative-times case. For clarity, let ${\displaystyle H=H_{\epsilon I,t=0}=H_{0}+gV}$ and ${\displaystyle U=U_{\epsilon I}(0,-\infty )}$

${\displaystyle i\hbar \epsilon g\partial _{g}\left(U|\Psi _{0}\rangle \right)=(H-E_{0})U|\Psi _{0}\rangle .}$

Now using the definition of ${\displaystyle \Psi _{\epsilon }}$ we differentiate and eliminate derivatives ${\displaystyle \partial _{g}(U|\Psi _{0}\rangle )}$ using the above expression, finding

${\displaystyle {\begin{aligned}i\hbar \epsilon g\partial _{g}|\Psi _{\epsilon }\rangle &={\frac {1}{\langle \Psi _{0}|U|\Psi _{0}\rangle }}(H-E_{0})U|\Psi _{0}\rangle -{\frac {U|\Psi _{0}\rangle }{{\langle \Psi _{0}|U|\Psi _{0}\rangle }^{2}}}\langle \Psi _{0}|(H-E_{0})U|\Psi _{0}\rangle \\&=(H-E_{0})|\Psi _{\epsilon }\rangle -|\Psi _{\epsilon }\rangle \langle \Psi _{0}|H-E_{0}|\Psi _{\epsilon }\rangle \\&=\left[H-E\right]|\Psi _{\epsilon }\rangle .\end{aligned}}}$

where ${\displaystyle E=E_{0}+\langle \Psi _{0}|H-H_{0}|\Psi _{\epsilon }\rangle }$. We can now let ${\displaystyle \epsilon \rightarrow 0^{+}}$ as by assumption the ${\displaystyle g\partial _{g}|\Psi _{\epsilon }\rangle }$ in left hand side is finite. We then clearly see that ${\displaystyle |\Psi _{\epsilon }\rangle }$ is an eigenstate of ${\displaystyle H}$ and the proof is complete.

References

1. Gell-Mann, Murray; Low, Francis (1951-10-15). "Bound States in Quantum Field Theory". Physical Review. 84 (2): 350–354. Bibcode:1951PhRv...84..350G. doi:10.1103/PhysRev.84.350. ISSN   0031-899X.
2. Hepp, Klaus (1969). Théorie de la renormalisation. Lecture Notes in Physics (in French). Vol. 2. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/bfb0108958. ISBN   978-3-540-04640-0.
3. G, Nenciu; G, Rasche (1989). "Adiabatic theorem and Gell-Mann-Low formula". Helvetica Physica Acta. 62: 372. doi:10.5169/SEALS-116035.
4. Molinari, Luca Guido (2007). "Another proof of Gell-Mann and Low's theorem". Journal of Mathematical Physics. 48 (5): 052113. arXiv: math-ph/0612030 . Bibcode:2007JMP....48e2113M. doi:10.1063/1.2740469. ISSN   0022-2488. S2CID   119665963.

• A.L. Fetter and J.D. Walecka: "Quantum Theory of Many-Particle Systems", McGraw–Hill (1971)