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| The '''Gell-Mann and Low theorem''' is a theorem in [[quantum field theory]] 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 [[Propagator (Quantum Theory)|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.
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| == History ==
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| The theorem was proved first by [[Gell-Mann]] and [[Francis E. Low|Low]] in 1951, making use of the [[Dyson series]]. In 1969 [[Klaus Hepp]] provided an alternative derivation for the case where the original Hamiltonian describes free particles and the interaction is norm bounded. In 1989 Nenciu and Rasche proved it using the [[adiabatic theorem]]. A proof that does not rely on the Dyson expansion was given in 2007 by Molinari. | | The stylish wardrobe of Maggie Gyllenhaal�s role in BBC Two�s hard-hitting political thriller, The Honourable Woman, has caught the attention of the eagle-eyed viewers.<br><br>The eight-part series, set against the backdrop of the Israeli-Palestinian conflict, centres around Nessa Stein, played by Gyllenhaal. Stein is an Anglo-Israeli businesswoman recently ennobled in the House of Lords who devotes herself to [http://philanthropicpurposes.org/ philanthropic purposes] across the Middle East, but hides a secret past from her time spent in Gaza eight years earlier.<br><br>Through the unravelling of her public and private life played out on an international, political stage, Stein parades in an increasingly impressive selection of outfits.<br>�Because the character of Nessa is so complicated and multi layered, we looked at all sorts of different people as reference. I suppose we started off by looking at other powerful and stylish women through history, Jackie Kennedy, Eva Peron, Margaret Thatcher, Cleopatra� Edward K Gibbon costume designer for the series [http://www.pcs-systems.co.uk/Images/celinebag.aspx http://www.pcs-systems.co.uk/Images/celinebag.aspx] told The Independent<br><br> |
| | Maggie Gyllenhaal The Honourable Woma<br> |
| | �And then we kind of threw all the reference away and started afresh. The way Maggie looked as Nessa was constantly evolving throughout the six month shoot.� |
| | The series opens with Nessa clad in a Roland Mouret power dress. Her day to day look is a sartorial dream with tailored suits by the likes of Stella McCartney, Acne, Escada, Pringle and vintage Chane<br> |
| | �Silk blouses and wide legged pants based on 1970s Yves Saint Laurent originals were created by Hilary Marschner� explains Gibb<br><br> |
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| == Statement of the theorem ==
| | Outerwear includes coats by Mulberry, vintage finds from Jil Sander and a 1980s Gieves and Hawkes men�s co<br>. |
| Let <math>|\Psi_0\rangle</math> be an eigenstate of <math>H_0</math> with energy <math>E_0</math> and let the 'interacting' Hamiltonian be <math>H=H_0 + gV</math>, where <math>g</math> is a coupling constant and <math>V</math> the interaction term. We define a Hamiltonian <math>H_\epsilon=H_0 + e^{-\epsilon |t|}gV</math> which effectively interpolates between <math>H</math> and <math>H_0</math> in the limit <math>\epsilon \rightarrow 0^+</math> and <math>|t|\rightarrow\infty</math>. Let <math>U_{\epsilon I}</math> denote the evolution operator in the [[interaction picture]]. The Gell-Mann and Low theorem asserts that if the limit as <math>\epsilon\rightarrow 0^+</math> of
| | Even curled up in her panic room at night she sports silk slips by haute couture Parisian lingerie designer Carine Gilson and London based lingerie label Bod<br>. |
| | In pictures: Nessa Stein's wardrobe in The Honourable Woma<br> |
| | Shoes are by Acne, Christian Louboutin and Celine. With bags from Mulberry and John Lewis. �Nessa's wardrobe runs the full gamut from designer, through High Street, Charity shops and bespoke pieces� says Gib<br><br> |
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| : <math> |\Psi^{(\pm)}_\epsilon \rangle = \frac{ U_{\epsilon I} (0,\pm\infty) |\Psi_0 \rangle}{\langle \Psi_0 | U_{\epsilon I}(0,\pm\infty)|\Psi_0\rangle} </math>
| | �The clothing is always the way in [to the character]� Gyllenhaal told WWD. �I never played a character that didn�t care about what they were wearing.� |
| | | The Honourable Woman continues tonight, BBC2 at 9pm. |
| exists, then <math>|\Psi^{(\pm)}_\epsilon \rangle</math> are eigenstates of <math>H</math>.
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| Note that 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.
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| == Proof ==
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| 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 <math>H_\epsilon</math> and let <math>g=e^{\epsilon \theta}</math>. From Schrödinger's equation for the time-evolution operator
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| : <math> i\hbar \partial_{t_1} U_\epsilon(t_1,t_2) = H_\epsilon(t_1) U_\epsilon(t_1,t_2)</math>
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| and the boundary condition that <math>U_\epsilon(t,t)=1</math> we can formally write
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| : <math>
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| 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).
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| </math>
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| Focus for the moment on the case <math>0\geq t_2\geq t_1</math>. Through a change of variables we can write
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| : <math>
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| U_\epsilon(t_1,t_2) = 1+ \frac{1}{i\hbar} \int_{\theta +t_2}^{\theta+t_1} dt' (H_0 + e^{\epsilon t'} V) U_\epsilon(t'-\theta,t_2).
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| </math>
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| We therefore have that
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| : <math>
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| \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).
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| </math>
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| This result can be combined with the Schrödinger equation and its adjoint
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| : <math> -i\hbar \partial_{t_1} U_\epsilon(t_2,t_1) = U_\epsilon(t_2,t_1) H_\epsilon(t_1) </math>
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| to obtain
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| : <math>
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| 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).
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| </math>
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| The corresponding equation between <math>H_{\epsilon I}, U_{\epsilon I}</math> is the same. It can be obtained by pre-multiplying both sides with <math>e^{i H_0 t_1/\hbar}</math>, post-multiplying with <math>e^{i H_0 t_2/\hbar}</math> and making use of
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| : <math>
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| U_{\epsilon I} (t_1,t_2) = e^{i H_0 t_1/\hbar} U_{\epsilon}(t_1,t_2) e^{-i H_0 t_2 /\hbar}.
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| </math>
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| The other case we are interested in, namely <math>t_2\geq t_1 \geq 0</math> 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
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| <math>t_{1,2}</math> have mixed signs). In summary, we obtain
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| : <math>
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| \left(H_{\epsilon, t=0}-E_0 \pm i \hbar \epsilon g \partial_g\right) U_{\epsilon I}(0,\pm\infty) |\Psi_0\rangle = 0.
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| </math>
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| We proceed for the negative-times case. Abbreviating the various operators for clarity
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| : <math>i \hbar \epsilon g \partial_g \left(U|\Psi_0\rangle\right) = (H_\epsilon-E_0)U|\Psi_0\rangle.</math>
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| Now using the definition of <math>\Psi_\epsilon</math> we differentiate and eliminate derivatives <math>\partial_g(U|\Psi_0\rangle)</math> using the above expression, finding
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| : <math>
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| \begin{align}
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| i \hbar \epsilon g \partial_g | \Psi_\epsilon \rangle &=
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| \frac{1}{\langle\Psi_0| U |\Psi_0 \rangle} (H_\epsilon-E_0) U|\Psi_0\rangle
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| - \frac{U|\Psi_0\rangle }{{\langle\Psi_0 |U| \Psi_0 \rangle}^2} \langle \Psi_0 | H_\epsilon-E_0 | \Psi_0\rangle \\
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| &= (H_\epsilon-E_0)|\Psi_\epsilon\rangle - |\Psi_\epsilon\rangle \langle \Psi_0 |H_\epsilon-E_0|\Psi_\epsilon\rangle \\
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| & = \left[ H_\epsilon - E \right] |\Psi_\epsilon\rangle.
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| \end{align}
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| </math>
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| where <math>E = E_0 + \langle\Psi_0 | H_\epsilon-H_0 | \Psi_\epsilon\rangle</math>. We can now let <math>\epsilon\rightarrow 0^+</math> as by assumption the left hand side is finite. We then clearly see that <math>|\Psi_\epsilon\rangle</math> is an eigenstate of <math>H</math> and the proof is complete.
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| == References ==
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| 1. M. Gell-Mann and F. Low: "Bound States in Quantum Field Theory", [http://link.aps.org/doi/10.1103/PhysRev.84.350 Phys. Rev. 84, 350 (1951)]
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| 2. K. Hepp: Lecture Notes in Physics (Springer-Verlag, New York, 1969), Vol. 2.
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| 3. G. Nenciu and G. Rasche: "Adiabatic theorem and Gell-Mann-Low formula", Helv. Phys. Acta 62, 372 (1989).
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| 4. L.G. Molinari: "Another proof of Gell-Mann and Low's theorem", [http://link.aip.org/link/doi/10.1063/1.2740469 J. Math. Phys. 48, 052113 (2007)]
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| 5. A.L. Fetter and J.D. Walecka: "Quantum Theory of Many-Particle Systems", McGraw–Hill (1971)
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| [[Category:Theorems in mathematical physics]]
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| [[Category:Quantum field theory]]
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The stylish wardrobe of Maggie Gyllenhaal�s role in BBC Two�s hard-hitting political thriller, The Honourable Woman, has caught the attention of the eagle-eyed viewers.
The eight-part series, set against the backdrop of the Israeli-Palestinian conflict, centres around Nessa Stein, played by Gyllenhaal. Stein is an Anglo-Israeli businesswoman recently ennobled in the House of Lords who devotes herself to philanthropic purposes across the Middle East, but hides a secret past from her time spent in Gaza eight years earlier.
Through the unravelling of her public and private life played out on an international, political stage, Stein parades in an increasingly impressive selection of outfits.
�Because the character of Nessa is so complicated and multi layered, we looked at all sorts of different people as reference. I suppose we started off by looking at other powerful and stylish women through history, Jackie Kennedy, Eva Peron, Margaret Thatcher, Cleopatra� Edward K Gibbon costume designer for the series http://www.pcs-systems.co.uk/Images/celinebag.aspx told The Independent
Maggie Gyllenhaal The Honourable Woma
�And then we kind of threw all the reference away and started afresh. The way Maggie looked as Nessa was constantly evolving throughout the six month shoot.�
The series opens with Nessa clad in a Roland Mouret power dress. Her day to day look is a sartorial dream with tailored suits by the likes of Stella McCartney, Acne, Escada, Pringle and vintage Chane
�Silk blouses and wide legged pants based on 1970s Yves Saint Laurent originals were created by Hilary Marschner� explains Gibb
Outerwear includes coats by Mulberry, vintage finds from Jil Sander and a 1980s Gieves and Hawkes men�s co
.
Even curled up in her panic room at night she sports silk slips by haute couture Parisian lingerie designer Carine Gilson and London based lingerie label Bod
.
In pictures: Nessa Stein's wardrobe in The Honourable Woma
Shoes are by Acne, Christian Louboutin and Celine. With bags from Mulberry and John Lewis. �Nessa's wardrobe runs the full gamut from designer, through High Street, Charity shops and bespoke pieces� says Gib
�The clothing is always the way in [to the character]� Gyllenhaal told WWD. �I never played a character that didn�t care about what they were wearing.�
The Honourable Woman continues tonight, BBC2 at 9pm.