View source for Physics:Schrödinger functional

{{Short description|Approach to quantum field theory}}
In [[Physics:Mathematical physics|mathematical physics]], some approaches to [[Physics:Quantum field theory|quantum field theory]] are more popular than others. For historical reasons, the Schrödinger representation is less favored than [[Physics:Fock space|Fock space]] methods. In the early days of [[Physics:Quantum field theory|quantum field theory]], maintaining symmetries such as [[Physics:Lorentz scalar|Lorentz invariance]], displaying them manifestly,  and proving renormalisation were of paramount importance. The Schrödinger representation is not manifestly Lorentz invariant and its renormalisability was only shown as recently as the 1980s by [[Biography:Kurt Symanzik|Kurt Symanzik]] (1981).

The '''Schrödinger functional''' is, in its most basic form, the time translation generator of state wavefunctionals. In layman's terms, it defines how a system of [[Physics:Quantum|quantum]] particles evolves through time and what the subsequent systems look like.

==Background==
Quantum mechanics is defined over the spatial coordinates <math>\mathbf{x}</math> upon which the Galilean group acts, and the corresponding operators act on its state as <math>\hat{\mathbf x}\psi(\mathbf{x})= \mathbf{x}\psi(\mathbf{x})</math>.  The state is characterized by a [[Physics:Canonical quantization|wave function]] <math>\psi(\mathbf{x})=\langle\mathbf{x}|\psi\rangle</math> obtained by projecting it onto the coordinate eigenstates defined by <math>\hat{\mathbf x}\left|\mathbf{x}\right\rangle = \mathbf{x}\left|\mathbf{x}\right\rangle</math>.  These eigenstates are not [[Physics:Stationary state|stationary]].  [[Time evolution]] is generated by the Hamiltonian, yielding the Schrödinger equation <math>i\partial_0\left|\psi(t)\right\rangle = \hat{H}\left|\psi(t)\right\rangle</math>.

However, in [[Physics:Quantum field theory|quantum field theory]], the coordinate is the field operator <math>\hat{\phi}_\mathbf{x}=\hat{\phi}(\mathbf{x})</math>, which acts on the state's wave functional as
:<math>\hat{\phi}(\mathbf{x}) \Psi\left[\phi(\cdot)\right] = \operatorname\phi\left(\mathbf{x}\right) \Psi\left[\phi(\cdot)\right] ,</math>
where "{{math|⋅}}" indicates an unbound spatial argument.  This wave functional
:<math>\Psi\left[\phi(\cdot)\right] = \left\langle\phi(\cdot)|\Psi\right\rangle</math>

is obtained by means of the field eigenstates 
:<math>\hat{\phi}(\mathbf{x}) \left|\Phi(\cdot)\right\rangle = \Phi(\mathbf{x}) \left|\Phi(\cdot)\right\rangle ,</math>
which are indexed by unapplied "classical field" configurations <math>\Phi(\cdot)</math>.  These eigenstates, like the [[Physics:Position operator#Eigenstates|position eigenstates]] above, are not [[Physics:Stationary state|stationary]].  Time evolution is generated by the [[Physics:Hamiltonian (quantum mechanics)|Hamiltonian]], yielding the Schrödinger equation,
:<math>i\partial_0\left|\Psi(t)\right\rangle = \hat{H}\left|\Psi(t)\right\rangle .</math>

Thus the state in quantum field theory is conceptually a functional superposition of field configurations.

==Example: scalar field==
In the [[Physics:Quantum field theory|quantum field theory]] of (as example) a quantum [[Scalar field|scalar field]] <math>\hat{\phi}(x)</math>, in complete analogy with the one-particle [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]], the eigenstate of this quantum field with the "classical field" <math>\phi(x)</math> ([[Physics:C-number|c-number]]) as its eigenvalue,
:<math>\hat{\phi}(x)\left|\phi\right\rangle =\phi\left(x\right)\left|\phi\right\rangle </math> 
is (Schwartz, 2013)
:<math>\left|\phi\right\rangle \propto e^{-\int dx\frac{1}{2}~(\phi(x)-\hat{\Phi}_{+}(x))^{2}}\left|0\right\rangle </math>
where <math>\hat{\Phi}_{+}\left(x\right)</math> is the part of <math>\hat{\phi}\left( x\right)</math>
that ''only includes [[Physics:Creation and annihilation operators|creation operators]]'' <math>a^\dagger_k</math>.  For the oscillator, this corresponds to the representation change/map to  the |''x''⟩ state from Fock states.

For a time-independent Hamiltonian {{mvar|H}}, the Schrödinger functional is defined as
:<math>\mathcal{S}[\phi_2,t_2;\phi_1,t_1]=\langle\,\phi_2\,|e^{-iH(t_2-t_1)/\hbar}|\,\phi_1\,\rangle.</math>

In the [[Physics:Schrödinger picture|Schrödinger representation]], this functional generates time translations of state wave functionals,  through
:<math>\Psi[\phi_2,t_2] = \int\!\mathcal{D}\phi_1\,\,\mathcal{S}[\phi_2,t_2;\phi_1,t_1]\Psi[\phi_1,t_1] .</math>

===States===

The normalized, [[Physics:Quantum vacuum state|vacuum state]], free field wave-functional is the Gaussian

:<math>\Psi_0[\phi] = \det{}^{\frac{1}{4}} \left(\frac{K}{\pi}\right)\; e^{-\frac{1}{2}\int d\vec{x} \int d\vec{y}\, \phi(\vec{x}) K(\vec{x},\vec{y}) \phi(\vec{y}) } = \det{}^{\frac{1}{4}}\left(\frac{K}{\pi}\right)\; e^{-\frac{1}{2}\phi\cdot K \cdot\phi},</math>

where the covariance ''K'' is

:<math>K(\vec{x},\vec{y}) = \int \frac{d^3 k}{(2\pi)^3} \omega_{\vec k}\,e^{i \vec{k}\cdot(\vec{x}-\vec{y})}.</math>

This is analogous to (the Fourier transform of) the product of each k-mode's ground state in the [[Physics:Continuum limit|continuum limit]], roughly (Hatfield 1992)

:<math>\Psi_0[\tilde\phi] = \lim_{\Delta k\to 0}\;\prod_{\vec k} \left(\frac{\omega_{\vec k}}{\pi}\right)^{\frac 1 4} e^{-\frac{1}{2}\omega_{\vec k}\tilde\phi(\vec k)^2 \frac{\Delta k^3}{(2\pi)^3}} \to \left(\prod_{\vec k} \left(\frac{\omega_{\vec k}}{\pi}\right)^{\frac 1 4}\right) e^{-\frac{1}{2}\int\frac{d^3k}{(2\pi)^3} \omega_{\vec k}\tilde\phi(|\vec k|)^2}.</math>

Each k-mode enters as an independent [[Physics:Quantum harmonic oscillator|quantum harmonic oscillator]]. One-particle states are obtained by exciting a single mode, and have the form,

:<math>\Psi[\phi] \propto \int d\vec{x} \int d\vec{y}\, \phi(\vec{x}) K(\vec{x},\vec{y}) f(\vec{y}) \Psi_0[\phi] = \phi\cdot K\cdot f\, e^{-\frac{1}{2}\phi\cdot K\cdot\phi} .</math>

For example, putting an excitation in <math>\vec{k}_1</math> yields (Hatfield 1992)

:<math>\Psi_1[\tilde\phi] = \left(\frac{2 \omega_{k_1}}{(2\pi)^3}\right)^{\frac 1 2} \tilde\phi(\vec{k}_1) \Psi_0[\tilde\phi]</math>
:<math>\Psi_1[\phi] = \left(\frac{2 \omega_{k_1}}{(2\pi)^3}\right)^{\frac 1 2} \int d^3 y\,e^{-i \vec{k}_1 \cdot \vec y}\phi(\vec y)\Psi_0[\phi] .</math>

(The factor of <math>(2\pi)^{-3/2}</math> stems from Hatfield's setting <math>\Delta k = 1</math>.)

==Example: fermion field==

For clarity, we consider a massless [[Physics:Spinor|Weyl–Majorana]] field <math>\hat\psi(x)</math> in 2D space in SO<sup>+</sup>(1, 1), but this solution generalizes to any massive [[Physics:Dirac fermion|Dirac]] [[Physics:Bispinor|bispinor]] in SO<sup>+</sup>(1, 3).  The configuration space consists of functionals <math>\Psi[u]</math> of anti-commuting [[Physics:Grassmann number|Grassmann-valued]] fields {{math|''u''(''x'')}}.  The effect of <math>\hat\psi(x)</math> is

:<math>\hat\psi(x)|\Psi\rangle = \frac{1}{\sqrt 2}\left(u(x) + \frac{\delta}{\delta u(x)}\right) |\Psi\rangle .</math>

== References==
* Brian Hatfield, ''Quantum Field Theory of Point Particles and Strings''. Addison Wesley Longman, 1992. See Chapter 10 "Free Fields in the Schrödinger Representation".
* I.V. Kanatchikov, "Precanonical Quantization and the Schrödinger Wave Functional." ''Phys. Lett. A'' '''283''' (2001) 25&ndash;36. Eprint [https://arxiv.org/abs/hep-th/0012084 arXiv:hep-th/0012084], 16 pages.
* R. Jackiw, "Schrödinger Picture for Boson and Fermion Quantum Field Theories." In ''Mathematical Quantum Field Theory and Related Topics: Proceedings of the 1987 Montréal Conference Held September 1–5, 1987'' (eds. J.S. Feldman and L.M. Rosen, American Mathematical Society 1988).
* H. Reinhardt, C. Feuchter, "On the Yang-Mills wave functional in Coulomb gauge." ''Phys. Rev. D'' '''71''' (2005) 105002. Eprint [https://arxiv.org/abs/hep-th/0408237 arXiv:hep-th/0408237], 9 pages.
* D.V. Long, G.M. Shore, "The Schrödinger Wave Functional and Vacuum States in Curved Spacetime." ''Nucl.Phys. B'' '''530''' (1998) 247&ndash;278. Eprint [https://arxiv.org/abs/hep-th/9605004 arXiv:hep-th/9605004], 41 pages.
* Kurt Symanzik, "Schrödinger representation and Casimir effect in renormalizable quantum field theory". ''Nucl. Phys.B'' '''190''' (1981) 1–44, [https://dx.doi.org/10.1016/0550-3213(81)90482-X doi:10.1016/0550-3213(81)90482-X].
* K. Symanzik, "Schrödinger Representation in Renormalizable Quantum Field Theory". Chapter in ''Structural Elements in Particle Physics and Statistical Mechanics'', NATO Advanced Study Institutes Series '''82''' (1983) pp 287–299, [https://dx.doi.org/10.1007/978-1-4613-3509-2_20 doi:10.1007/978-1-4613-3509-2_20].
* Martin Lüscher, Rajamani Narayanan, Peter Weisz, Ulli Wolff, "The Schrödinger Functional - a Renormalizable Probe for Non-Abelian Gauge Theories". ''Nucl.Phys.B'' '''384''' (1992) 168–228, [https://dx.doi.org/10.1016/0550-3213(92)90466-O doi:10.1016/0550-3213(92)90466-O]. Eprint [https://arxiv.org/abs/hep-lat/9207009 arXiv:hep-lat/9207009].
* Matthew  Schwartz (2013). ''Quantum Field Theory and the Standard Model'', Cambridge University Press, Ch.14.

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[[Category:Quantum field theory]]

{{Sourceattribution|Schrödinger functional}}