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Field Formulation of Many-Body Quantum Physics

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Field Formulation of Many-Body Quantum Physics A common mistake that people make when trying to design something completely foolproof is to underestimate the ingenuity of complete fools. Douglas Adams (1952-2001) 2 Field Formulation of Many-Body Quantum Physics A piece of matter composed of a large number of microscopic particles is called a many-body system. The microscopic particles may either be all identical or of dif- ferent species. Examples are crystal lattices, liquids, and gases, all of these being aggregates of molecules and atoms. Molecules are composed of atoms which, in turn, consist of an atomic nucleus and electrons, held together by electromagnetic forces, or more precisely their quanta, the photons. The mass of an atom is mostly due to the nucleus, only a small fraction being due to the electrons and an even smaller fraction due to the electromagnetic binding energy. Atomic nuclei are themselves bound states of nucleons, held together by mesonic forces, or more precisely their quanta, the mesons. The nucleons and mesons, finally, consist of the presently most fundamental objects of nuclear material, called quarks, held together by gluonic forces. Quarks and gluons are apparently as fundamental as electrons and photons. It is a wonderful miracle of nature that this deep hierarchy of increasingly funda- mental particles allows a common description with the help of a single theoretical structure called quantum field theory. As a first step towards developing this powerful theory we shall start from the well-founded Schr¨odinger theory of nonrelativistic spinless particles. We show that there exists a completely equivalent formulation of this theory in terms of quantum fields. This formulation will serve as a basis for setting up various quantum field theoretical models which can eventually explain the physics of the entire particle hierarchy described above. 2.1 Mechanics and Quantum Mechanics for n Distinguishable Nonrelativistic Particles For a many-body system with only one type of nonrelativistic spinless particles of mass M, which may be spherical atoms or molecules, the classical Lagrangian has the form n M L(x , x˙ ; t)= x˙ 2 V (x ,..., x ; t), (2.1) ν ν 2 ν − 1 n νX=1 82 2.1 Mechanics and Quantum Mechanics for n Nonrelativistic Particles 83 where the arguments xν , x˙ ν in L(xν, x˙ ν; t) stand, pars pro toto, for all positions xν and velocities x˙ ν , ν = 1,...,n. The general n-body potential V (x1,..., xn; t) can usually be assumed to consist of a sum of an external potential V1(xν; t) and a pair potential V (x x ; t), also called one- and two-body-potentials, respectively: 2 ν − µ 1 V (x1,..., xn; t)= V1(xν; t)+ V2(xν xµ; t). (2.2) ν 2 ν,µ − X X The second sum is symmetric in µ and ν, so that V (x x ; t) may be taken as 2 ν − µ a symmetric function of the two spatial arguments — any asymmetric part would not contribute. The symmetry ensures the validity of Newton’s third law “actio est reactio”. The two-body potential is initially defined only for µ = ν, and the sum is 6 restricted accordingly, but for the development to come it will be useful to include also the µ = ν -terms into the second sum (2.2), and compensate this by an appro- priate modification of the one-body potential V1(xν; t), so that the total potential re- mains the same. Such a rearrangement excludes pair potentials V2(xν xµ; t) which are singular at the origin, such as the Coulomb potential between− point charges 2 V2(xν xµ; t) = e /4π xν xµ . Physically, this is not a serious obstacle since all charges− in nature really| − have| a finite charge radius. Even the light fundamen- tal particles electrons and muons possess a finite charge radius, as will be seen in Chapter 12. At first we shall consider all particles to be distinguishable. This assumption is often unphysical and will be removed later. For the particle at xν, the Euler- Lagrange equation of motion that extremizes the above Lagrangian (2.1) reads [recall Eq. (1.8)] ∂V1(xν; t) ∂ Mx¨ν = V2(xν xµ; t). (2.3) − ∂xν − µ ∂xν − X The transition to the Hamiltonian formalism proceeds by introducing the canonical momenta [see (1.10)] ∂L pν = = Mx˙ ν , (2.4) ∂x˙ ν and forming the Legendre transform [see (1.9)] H(pν, xν; t) = pνx˙ ν L(xν , x˙ ν; t) " ν − # X x˙ ν =pν /M 2 pν 1 = + V1(xν; t)+ V2(xν xµ; t). (2.5) ν 2M ν 2 ν,µ − X X X From this, the Hamilton equations of motions are derived as [see (1.17)] ∂V1(xν ; t) ∂ p˙ ν = H, pν = V2(xν xµ; t), (2.6) { } − ∂xν − µ ∂xν − X ∂H pν x˙ ν = H, xν = = , (2.7) { } ∂pν M 84 2 Field Formulation of Many-Body Quantum Physics with A, B denoting the Poisson brackets defined in Eq. (1.20): { } n ∂A ∂B ∂B ∂A A, B = . (2.8) { } ∂pν ∂xν − ∂pν ∂xν ! νX=1 An arbitrary observable F (pν, xν; t) changes as a function of time according to the equation of motion (1.19): dF ∂F = H, F + . (2.9) dt { } ∂t It is now straightforward to write down the laws of quantum mechanics for the system. We follow the rules in Eqs. (1.236)–(1.238), and take the local basis x ,..., x (2.10) | 1 ni as eigenstates of the position operators xˆν : xˆ x ,..., x = x x ,..., x , ν =1, . , n. (2.11) ν | 1 ni ν| 1 ni They are orthonormal to each other: x ,..., x x′ ,..., x′ = δ(3)(x x′ ) δ(3)(x x′ ), (2.12) h 1 n| 1 ni 1 − 1 ··· n − n and form a complete basis in the space of localized n-particle states: 3 3 d x1 d xn x1,..., xn x1,..., xn =1. (2.13) Z ··· | ih | An arbitrary state is denoted by a ket vector and can be expanded in this basis by multiplying the state formally with the unit operator (2.13), yielding the expansion 3 3 ψ(t) 1 ψ(t) = d x1 d xn x1,..., xn x1,..., xn ψ(t) . (2.14) | i ≡ ×| i Z ··· | ih | i The scalar products x ,..., x ψ(t) ψ(x ,..., x ; t) (2.15) h 1 n| i ≡ 1 n are the wave functions of the n-body system. They are the probability amplitudes for the particle 1 to be found at x1, particle 2 at x2, etc. The Schr¨odinger equation reads, in operator form, Hˆ ψ(t) = H(pˆ , xˆ ; t) ψ(t) = ih∂¯ ψ(t) , (2.16) | i ν ν | i t| i where pˆν are Schr¨odinger’s momentum operators, whose action upon the wave func- tions is specified by the rule x ,..., x pˆ = ih∂¯ x x ,..., x . (2.17) h 1 n| ν − ν h 1 n| 2.2 Identical Particles: Bosons and Fermions 85 Multiplication of (2.16) from the left with the bra vectors x1,..., xn yields via (2.17) the Schr¨odinger differential equation for the wave functioh ns: | H( ih∂¯ xν , xν; t)ψ(x1,..., xn; t) − 2 h¯ 2 1 = ∂xν + V1(xν; t)+ V2(xν xµ; t) ψ(x1,..., xn; t) "− ν 2M ν 2 ν,µ − # X X X = ih∂¯ tψ(x1,..., xn; t). (2.18) In many physical systems, the potentials V (x ; t) and V (x x ; t) are independent 1 ν 2 ν − µ of time. Then we can factor out the time dependence of the wave functions with fixed energy as −iEt/¯h ψ(x1,..., xn; t)= e ψE(x1,..., xn), (2.19) and find for ψE(x1,..., xn) the time-independent Schr¨odinger equation H( ih∂¯ x , x ; t)ψ (x ,..., x )= Eψ (x ,..., x ). (2.20) − ν ν E 1 n E 1 n 2.2 Identical Particles: Bosons and Fermions The quantum mechanical rules in the last section were written down in the previ- ous section under the assumption that all particles are distinguishable. For most realistic n-body systems, however, this is an unphysical assumption. For example, there is no way of distinguishing the electrons in an atom. Thus not all of the solutions ψ(x1,..., xn; t) of the Schr¨odinger equation (2.18) can be physically per- missible. Consider the case where the system contains only one species of identical particles, for example electrons. The Hamilton operator in Eq. (2.18) is invariant under all permutations of the particle labels ν. In addition, all probability ampli- tudes ψ ψ must reflect this invariance. They must form a representation space h 1| 2i of all permutations. Let Tij be an operator which interchanges the variables xi and xj. T ψ(x ,..., x ,..., x ,..., x ; t) ψ(x ,..., x ,..., x ,..., x ; t). (2.21) ij 1 i j n ≡ 1 j i n It is called a transposition. The invariance may then be expressed as T ψ T ψ = h ij 1| ij 2i ψ1 ψ2 , implying that the wave functions ψ1 and ψ2 can change at most by a phase:h | i | i | i iαij Tijψ(x1,..., xn; t)= e ψ(x1,..., xn; t). (2.22) But from the definition (2.21) we see that Tij satisfies 2 Tij =1. (2.23) Thus eiαij can only have the values +1 or 1. Moreover, due to the identity of all − particles, the sign must be the same for any pair ij. The set of all multiparticle wave functions can be decomposed into wave functions transforming in specific ways under arbitrary permutations P of the coordinates.
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