Week 1 Lecture: Concepts of Quantum Field Theory (QFT)

Transcription

1 Wee 1 Lecture: Concepts of Quantum Field Theory QFT Andrew Forrester April 4, 008 Relative Wave-Functional Probabilities This Wee s Questions What are the eact solutions for the Klein-Gordon field? What about for two Klein-Gordon-coupled oscillators? What about for a lattice of Klein-Gordon-coupled oscillators? Are the energy eigen-qft-states also configuration eigenstates of the field? Might the energy eigen-qft-states actually be modes of the collective wavefunctions, rather than modes of the collective point-lie oscillators? This would mean that for a given energy there is only a high probability that the field taes on a planewave configuration. Does the Hamiltonian energy operator H act on fields or statefunctions? What is the intuitive version of Ψ 0 ϕ? d 3 p 3 ω p a p a p + 1 a p, a p Is the wave-functional ratio Ψ 0 ϕ/ψ 0 0 the relative probability density that the ground state field will be measured as or will interact as ϕ? According to Kanatchiov, it is. 1

2 Important Information Gained from the Literature Kanatchiov An important claim is made by Kanatchiov in the first paragraph of section 4:... the Schrödinger wave functional Ψ S y a, t... is nown to be the probability amplitude of observing the field configuration y a on a space-lie hypersurface of constant time t. Kanatchiov also proposes the intuitive wave-functional formula that I propose! See comments in the References. Jaciw Some important comments are made by Jaciw 4 shown in the References below, along with the following. I ve condensed and slightly changed the material to suit my notational conventions. States are viewed as functionals Ψϕ of a c-number field at fied time ϕ. An inner product is defined by functional integration. Ψ 1 Ψ Dϕ Ψ 1ϕ Ψ ϕ Ψ Ψ 1. The operators Φ and Π act on states as follows Φ Ψϕ ϕ Ψϕ, Π Ψϕ δ i δϕ Ψϕ. While the analogy with quantum mechanics is obvious, a difference emerges when we consider a Foc basis for our functional space. This basis is the field theoretic analog of a harmonic oscillator basis in quantum mechanics. A Foc vacuum is a Gaussian functional with covariance Ω, which is symmetric and can be comple, but possesses a positive definite real part Ψ Ω ϕ det 1/4 Ωr Ω Ω r + iω i, ep 1 ϕωϕ, Ω, y Ωy,. An obvious functional notation is used throughout 1, ϕωϕ d dy ϕ Ω, y ϕy, and the determinant is functional. In spite of the possibility for greater generality, we shall always use translational invariant covariances, dp Ω, y e ip y Ωp, which are diagonal in the momentum representation. d dy e ip Ω, y e iqy Ωp δp q. Higher basis states are polynomials in ϕ multiplying Ψ Ω, and they are orthonormalized if linear combinations corresponding to functional Hermite polynomials are taen. This defines a Foc space within our functional space. 1 Note: This loos close to the integral I obtained in the Bonus Lecture from last quarter in Attempt R 1 ϕ 1 ϕ.

3 In our functional space, different Foc bases constructed with different covariances can be inequivalent. This happens because field theory possesses an infinite number of degrees of freedom. For consider two Foc vacua with covariances Ω 1 and Ω. Their overlap is e N Ω 1 Ω. For eample when the covariances are real, N is given by N L d ln 1 Ω 1 Ω + Ω, Ω 1 where L is the length of the space. Note: I m not sure what equation.7 means. The Intuitive Wave-Functional of the Fourier-Transformed Field The state-function in terms of a partial Fourier transform and a partial Fourier transform in terms of the usual state-function are Ψ Ψ, φ d 3 p 3/ Ψp, φ eip Ψp, φ The full Fourier transform of the state-function is Ψ Ψp, Ψp,. 3/ Ψ, φ e ip. The field and its Fourier transform are ϕ ϕ 3/ ϕp e ip ϕ ϕp ϕp 3/ ϕ e ip. So, let s try to epress the wave-functional Ψϕ Ψ, ϕ e ln Ψ,ϕ e P ln Ψ,ϕ er ln Ψ,ϕ in terms of the Fourier-transformed field: Ψ ϕ Ψ, ϕ Ψ, 3/ d 3 p 3/ Ψ p, ep ln ϕp e ip 3/ d 3 p 3/ Ψ p, ϕp e ip 3/ e ip ϕp e ip e ip p? It d be nice if it were something lie p Ψ p, ϕp 3

4 Graphing the Wave-Functional on the Field Fourier Transform I solved this initially last quarter in my hand-written notes for Lecture 9 of Winter 008. The wave-functional Ψ 0 ϕ on the field Fourier transform ϕ is given in Hatfield 1 Eqn as Ψ 0 ϕ ep 1 d m ϕ ϕ using ω + m, given at the top of that page pg 03. Let s assume that the wave-functional ratio Ψ 0 ϕ/ψ 0 0 is the relative probability density that the ground state Fourier transformed field will be measured as or will interact as ϕ. Q: How liely is it that the ground state manifests itself as a planewave eigen-field ϕ q? A: Just as liely as it manifests itself as being completely flat. Ecept for a planewave that is 1 everywhere. ϕ q Re e iq cosq Ψ 0 ϕ q /Ψ 0 0 ep ϕ q p ep ep ϕ q e iq 3/ ϕ q e ip eiq p 3/ 3/ δq p d m 3/ δq 3/ δq + d 3 + m δq δq q + m δq { e 0 1 q 0 impossible q 0 Q: How liely is it that the ground state manifests itself as a Gaussian eigen-field ϕ G? A: Some Gaussians are equally liely, some are less liely, some are more even infinitely more liely. ϕ G 1 3 ϕ G σ e /σ a 3/ e a a 1/σ 4

5 ϕ G p 3/ ϕ G e ip a 3/ ip 3/ e a a 3/ +ip /a 3/ e a a 3/ 3/ e a+ip/a e p /4a a 3/ e p /4a 1 3/ d 3 u e au a 3/ e p /4a 1 3/ 3/ a 1 /4a 3/ e p 3/ e σ p / Ψ 0 ϕ G /Ψ 0 0 ep 1 ep 1 d m 3/ e σ / 3/ e σ / d 3 + m e σ Mathematica doesn t now how to solve this 3-D version, so let s go to 1-D: Ψ 0 ϕ G /Ψ 0 0 ep 1 d + m 1/ e σ / 1/ e σ / ep 1 d + m e σ ep σ HypergeometricU 1, 0, σ m I solved and graphed this using Mathematica. There are many points σ, m where this epression is greater than 1 and approaches infinity and where it s less than 1. Comment: This does not follow my intuition. I would epect the flat ϕ 0 field to be the most liely configuration. Planewaves and Gaussians should not be equally liely and should not be more liely. 5

6 References 1 Brian Hatfield: Quantum Field Theory of Point Particles and Strings, Addison Wesley Longman, Inc. 199 Chapter 10: Free Fields in the Schrödinger Representation Igor V. Kanatchiov: Precanonical quantization and the Schrdinger wave functional, Physics Letters A Volume 83, Issues 1-, 7 May 001, Pages 5-36 This is the first paper that I ve found that contains the intuitive formula for the wave-functional. In Kanatchiov s notation where Σ is a Cauchy surface, etc.: Ψ y a, t Σ Ψ Σ Ψ y a,, t Ψ y a,, t e P ln Ψya,,t Additional statements are made that I have not made or verified, such as e P ln R Ψya,,t lim d ln Ψy a,,t 0 e1/ 3 Also, an important claim is made in the first paragraph of section 4:... the Schrödinger wave functional Ψ S y a, t... is nown to be the probability amplitude of observing the field configuration y a on a space-lie hypersurface of constant time t. This is what I ve been assuming in my calculations in this lecture and prior lectures. And this is an answer to one of the questions I ased this wee. 3 jostpuur: wave functional Physics Forum Thread < inde.php/t html > Accessed 03 April 008 This person seems to have questions similar to ours. 4 R. Jaciw: Schrödinger Picture for Boson and Fermion Quantum Field Theories, Canadian Mathematical Society CMS Conference Proceedings, Volume 9; Mathematical Quantum Field Theory and Related Topics: Proceedings of the 1987 Montréal Conference held September 1-5, 1987; Published by the American Mathematical Society 1988 I forgot to list this boo in a previous lecture, but this is only the second source I ve found that introduces the Schrödinger picture in any depth. Important: The Schrödinger picture approach gains a unique advantage when analyzing timedependent problems, lie field theory in de Sitter space or out of thermal equilibrium, where the concept of Foc vacuum is ill-defined. It also has a stylistic advantage over the conventional approach because one can well-define renormalized generators without normal ordering with respect to a pre-selected vacuum. 6