Many particle physics as a quantum field theory

Transcription

1 Many particle physics as a quantum field theory Infobox 0.1 Summary of this chapter In a quantum mechanical system of N identical particles, the wavefunction, ψ σ1...σ N ( x 1,..., x N, t), is a function of time, t, the positions, x 1,..., x N, and the spins σ 1... σ N of the particles. It must solve the Schrödinger equation, i h t ψ σ 1...σ N ( x 1,..., x N, t) = + i<j J ρ i ρ j =1 ρ iρ j N i=1 h 2 2 i 2m ψ σ 1...σ N ( x 1,..., x N, t) Vσ i σ j ( x i x j )ψ σ1...ρ i...ρ j...σ N ( x 1,..., x N, t) with appropriate boundary conditions. For a spin-independent two-body potential V ρρ ( x y) = δ ρ v( x y) σσ σ δ ρ σ The wave-function is either completely symmetric under permutations of the labels of spins and positions σ 1... σ N ; x 1... x N in the case of bosons, or completely anti-symmetric, in the case of fermions. Wave-functions should be normalized, d 3 x 1... d 3 x N J σ 1...σ N =1 ψ σ 1...σ N ( x 1,..., x N, t)ψ σ1...σ N ( x 1,..., x N, t) = 1 (or the appropriate generalization for continuum normalization). The equivalent second quantized theory in the Schrödinger picture has the state vector Ψ(t) > obeying the Schrödinger equation and being an eigenstate of the number operator with eigenvalue N, i h Ψ(t) >= H Ψ(t) >, N Ψ(t) >= N Ψ(t) > t 1

2 where the number and hamiltonian operators are N = H = d 3 x d 3 x 2 ψ σ ( x)ψ σ ( x) σ=1 2 h 2 ψ 2m σ ( x) ψ σ ( x) σ= d 3 xd 3 y J σσ ρρ =1 ψ σ ( x)ψ σ ( y)v ρρ σσ ( x y)ψ ρ ( y)ψ ρ ( x) Here, σ, ρ label spin states. The quantized fields obey the commutation or anti-commutation relations: ψσ ( x), ψ ρ ( y) ] = δρ σδ( x y) ψ σ ( x), ψ ρ ( y)] = 0, ψ σ ( x), ψ ρ ( y) ] = 0 The commutation relations A, B] AB BA are for bosons and the anti-commutation relations A, B] + AB + BA are for fermions. The second quantized theory in the Heisenberg picture has the field equation: ) h2 (i h + t 2m 2 ψ σ ( x, t) = d 3 yv ρρ σσ ( x y)ψ σ ( y, t)ψ ρ ( y, t)ψ ρ ( x, t) and the equal-time commutation relations ψσ ( x, t), ψ ρ ( y, t) ] = δρ σδ( x y), ψ σ ( x, t), ψ ρ ( y, t)] = 0, ψ σ ( x, t), ψ ρ ( y, t) ] = 0 The wave equation and equal time commutation relations above are a definition of the many-particle problem which is closest in spirit to a quantum field theory. 2

3 1 Introduction In this chapter we will attempt to develop intuition for the answer to the question what is a quantum field theory. We will do this by studying a system with many particles. For now, we will assume that the particles are non-relativistic. The generalization to relativistic particles and relativistic quantum field theory will be discussed later. We will assume that the problem in front of us is quantum mechanical, that is, that we want to find a solution of the Schrödinger equation for the system as a whole and then use that solution, the wave function, to answer questions about the physical attributes of the system. In order to describe the quantum mechanical problem for a large number of particles in an elegant way, we will develop a procedure which is called second quantization. Many researchers dislike this term and dismiss it with the statement that there is no first or second or third quantization, there is only quantization. This is an accurate statement. Our second quantization is not another type of quantization. We emphasize that it is merely a re-writing of the same old quantization, albeit using different tools. In non-relativistic quantum mechanics, when the total number of particles is finite, second quantization gives an alternative, but at the same time completely equivalent formulation of the problem of solving the Schrödinger equation. This formulation is convenient for some applications, such as perturbation theory which is widely used to study many-particle systems. These can be relevant to many interesting physical scenarios. Metals, superconductors, superfluids, and nuclear matter are important examples 1, 2, 3]. The formalism is particularly useful in that it allows us to take the thermodynamic limit which is an idealization of such a system that considers the limit as both the volume of the system and the total number of particles in the system go to infinity, with the density the number of particles per unit of volume kept finite. The system can simplify somewhat in that limit. Moreover, it can be a good approximation to real systems, where the number of particles in a macroscopic system is typically very large, of order Avogadro s number, and where the details of the boundaries of the system are not very important for the bulk properties which we attempt to describe. Later, we will generalize the second quantized system that we find in order to make it relativistic, that is, so that it can describe particles with velocities approaching the velocity of light. In this generalization, the analog of second quantization is essential. Relativistic quantum mechanics is necessarily a 3

4 many-particle theory and the number of particles is always infinite, so there is no convenient description of it using a many-particle Schrödinger equation. In both the relativistic and non-relativistic cases, the second quantized theory is a quantum field theory, that is, a quantum theory where fields are the dynamical variables. In classical field theory, a field is simply a function of space and time coordinates whose value at a given time and point in space has a physical interpretation. A familiar example of a classical field theory is classical electrodynamics where the electric field and magnetic field are the classical fields. We can think of classical electrodynamics as a mechanical theory where the dynamical variables are these classical fields and the mechanical problem is to determine the time evolution of the dynamical variables, in this case, to determine the electric and magnetic fields as functions of the space and time coordinates by solving Maxwell s equations. In the framework of classical electrodynamics, the electric field and magnetic field are vector-valued functions of the space and time coordinates. In a quantum field theory, instead of being ordinary functions, like the electric and magnetic fields which are studied in classical electrodynamics, the fields in a quantum field theory are space and time-dependent operators which act on vectors in a Hilbert space, the space of possible quantum states of the quantum field theory. In such a theory, the physical entities, those attributes which can be measured by doing experiments, for example, are the expectation values and correlations of various operators. We will get a more precise picture of how this works later. 2 Non-relativistic particles We will begin with the non-relativistic quantum mechanics of a system of particles which interact with each other through a two-body force. We will assume that the force between any two particles can be derived by taking the gradient of a potential energy, V ( x i, x j ), which is a function of the positions of the two particles. We say that V ( x i, x j ) is the potential energy which is stored in the interaction between particle i and particle j when particle i occupies position x i and particle j occupies position x j. The force on particle i when it is located at position x i due to the presence of particle j located at position x j is given by F ij = i V ( x i, x j ) 4

5 If the particles are identical, so that the energy of their configuration is not changed by interchanging the particle positions, the two-body potential must be a symmetric function, V ( x i, x j ) = V ( x j, x i ) We will assume that this is always the case. We will generally assume that these potentials do not depend on the time. We will also generally assume that they are functions of relative positions of the particles so that V ( x i, x j ) = V ( x i x j ) When this is the case, the forces between the particles obey Newton s first law, which states that the force of j on i is of equal magnitude and opposite direction to the force of i on j, F ij = F ji. Then, will generally also assume that the total potential energy of the N-particle system can be written as a sum of the potential energies which are stored in the interaction between the two members of each pair of particles. That is, the potential energy of an assembly of N particles, occupying positions labeled by coordinates x 1,..., x N is given by the function V( x 1,..., x N ) = N i<j=1 V ( x i, x j ) (1) The restriction i < j in the summation in (1) ensures that each pair of particle is counted exactly once. The total force on the i th particle due to the presence of the rest is gotten by taking the gradient F i = i V( x 1,..., x N ) = i N j=1,j i V ( x i x j ) ( where the gradient operator has the usual definition: i x 1 i ),,. x 2 i x 3 i An example of a force is the Coulomb interaction between two charged particles. If they have the same electric charge, e, the Coulomb potential energy is V ( x i, x j ) = 5 e 2 4π x i x j

6 Moreover, it depends only of the difference of the position vectors, x i x j, rather than depending on the positions individually. This has the physical meaning that the Coulomb energy of two particles depends only on their relative locations, and, with fixed relative locations, it does not depend on where the pair is located. This property of an interaction is called translation invariance and we shall see that it has important consequences for the quantum mechanical theory. Let us begin by examining the non-relativistic quantum mechanics of a system of N identical particles. The Hamiltonian for such a system is given by the sum of the kinetic energies of each particle and the interaction energy. It has the form H( p 1,..., p N, x 1,..., x N ) = N i=1 p 2 i 2m + N i<j=1 V ( x i x j ) (2) Here, x i is the position and p i is momentum of the i th particle and the index i runs over the labels of the particles, i = 1, 2,..., N. In the first term in the Hamiltonian, p 2 i /2m is the kinetic energy of the i th particle, and the first term is the sum of all of the kinetic energies of the particles. The second term in the Hamiltonian is the total potential energy that is stored in the interactions between the particles. The expression (2) is thus the total energy of a system of particles which have positions { x 1,..., x N } and momenta { p 1,..., p N }. The fact that the particles are identical is evident in the Hamiltonian where the particles have the same masses, m, and the two-body potentials are symmetric, V ( x i x j ) = V ( x j x i ). In the quantum mechanics of non-relativistic particles, the positions and momenta are operators. We will temporarily denote operators with a hat, so that they are { ˆx 1,..., ˆx N ˆp1,..., ˆp N }. The precise property that defines them as operators are the commutation relations ] ] a ˆx i, ˆx b j = 0, a ˆx i, ˆp b j = i hδij δ ab, ] a ˆp i, ˆp b j = 0 (3) There the labels i, j take values in the set {1, 2,..., N} and they label the distinct particles. The indices a, b take the values {1, 2, 3} and they label the three Cartesian components of the position or momentum vector of each particle. The right-hand-side of the non-zero commutation relations contains Planck s constant, h. The Hamiltonian in equation (2) is a function of positions and momenta. If positions and momenta become operators, the Hamiltonian also becomes 6

7 an operator by virtue of the fact that it depends on the positions and momenta. Then, we have Ĥ H( ˆp 1,..., ˆp N, ˆx 1,..., ˆx N ) = 2 N ˆpi 2m + i=1 N i<j=1 V ( ˆx i ˆx j ) (4) which defines Ĥ as an operator. In addition to the finding Hamiltonian operator, we shall also need the state-vector, ψ(t), which specifies the quantum state of the system. The operators {ˆx 1,..., ˆx N ˆp 1,..., ˆp N } act on state vectors. There is a precise sense in which we can think of the operators as matrices and the state-vector as a column vector. The operation of an operator on the state vector is then viewed as simple multiplication of a vector by a matrix. The dynamics of the quantum mechanical system is encoded in the time-dependence of the state vector. This tells you how the quantum state evolves from some initial state it its state at some value of the time parameter, t. This evolution is governed by the fact that the state vector should solve the Schrödinger equation, which in abstract form, is i h ψ(t) = Ĥψ(t) (5) t It is necessary to find a workable representation of the commutation relations (3) between the position and momentum operators. A common way to do this is to think of the operators ˆx a i and ˆp a i as operating on functions of all of the coordinates, f( x 1,..., x N ), with the operation of ˆx a i being simply the multiplication of the function by the variable x a i and the operation of ˆp a i as proportional to the partial derivative by x a i, ˆx a i f(t, x 1,..., x N ) = x a i f(t, x 1,..., x N ) (6) ˆp a i f(t, x 1,..., x N ) = h f(t, x 1,..., x N ) (7) i It is easy to see that this definition reproduces the commutation relation for position and momentum, x a i ˆx a i ˆp b j f(t, x 1,..., x N ) ˆp b j ˆx a i f(t, x 1,..., x N ) = x a h i f(t, x i x b 1,..., x N ) h x a i f(t, x 1,..., x N ) j i x b j = i hδ ij δ ab ]f(t, x 1,..., x N ) (8) 7

8 In this way of representing the operators, the state-vector is the wavefunction as we have been employing it, a function of the coordinates and the time, ψ( x 1,..., x N, t). The state-vector should solve the Schrödinger equation, which, with our explicit realization of the commutation relations by coordinates and derivatives, has the form of the linear partial differential equation i h t ψ( x N h 2 2 1,..., x N, t) = i 2m + ] V ( x i x j ) ψ( x 1,..., x N, t) (9) i=1 i<j where the index i on the Laplacian 2 i means that the derivatives are by the components of x i. We can present the Schrödinger equation (9) as a time-independent equation by making the ansatz ψ( x 1,..., x N, t) = e iet/ h ψ E ( x 1,..., x N ) Then (9) implies Eψ E ( x 1,..., x N ) = N i=1 h 2 2 i 2m + i<j V ( x i x j ) ] ψ E ( x 1,..., x N ) (10) The solution of this equation with boundary conditions should give us the wave-functions and the energies, E of stationary states. Here, ψ E ( x 1,..., x N ) is called an eigenstate or eigenvector of the Hamiltonian and E is the eigenvalue which is associated with it. The time-dependent wave-function should be normalized, d 3 x 1... d 3 x N ψ( x 1,..., x N, t) 2 = 1 for all values of the time, t. Also, wave-functions with different energy eigenvalues are orthogonal, d 3 x 1... d 3 x N ψ E ( x 1,..., x N )ψ E ( x 1,..., x N ) = δ EE Generally, equation (10) is difficult to solve when the interaction potential is non-trivial. In fact, there are very few examples of interaction potentials 8

9 where one can solve for the wave-functions or the energies exactly. One of them is the case of free particles, when the potential is zero. In that case, the explicit wave-function can be found, it is simply constructed from planewaves, ψ E ( x 1,..., x N ) = and the energy eigenvalue is 1 (2π h) 3 2 e i N i=1 k i x i / h E = N i=1 k 2 i 2m If the initial state, say at time t = 0 is given by a function ψ 0 ( x 1,..., x N ), the wave-function at any time is given by ψ( x 1,..., x N, t) = N j=1 d 3 k j d 3 y j (2π h) 3 k 2 j e i 2m t/ h+i k ( x j y j )/ h ] ψ 0σ1...σ N ( y 1,..., y N ) In fact, in this simple case, the integrations over k j can be done to get N ] d 3 y j ψ( x 1,..., x N, t) = e i m x j y j 2 (2π h 2 t/im) 3 2 ht ψ 0 ( y 1,..., y N ) 2 j=1 2.1 Identical particles There is one important aspect of the problem which we have ignored until now and which must be discussed here. We have constructed the Hamiltonian so that the particles are identical. They have the same masses and the interaction between any pair of particles is the same as the interaction between any other pair of particles. The Hamiltonian is unchanged if we trade the labels on the indices of the particles. That is, if we make the substitution { x 1, x 2,..., x N ; p 1, p 1,..., p N } { x P (1), x P (2),..., x P (N) ; p P (1), p P (2),..., p P (N) } where the permutation {1, 2,..., N} {P (1), P (2),..., P (N)} is simply a re-ordering of the integers {1, 2,..., N}. There are N! different possible permutations, including the identity. We require that the permutation act on the indices of both the particles and the momenta. This guarantees that the commutation relations (7) are also left unchanged by the transformation. 9

10 This permutation symmetry of the Hamiltonian has an important consequence. Consider the Schrödinger equation (10) and let us assume that we manage to solve the equation to find an allowed value of the energy, E, and the wave-function which corresponds to it, ψ E ( x 1,..., x N ). The permutation symmetry then tells us that ψ E ( x P (1),..., x P (N) ) also obeys the same equation, (10), for any of the N! distinct permutations. What is more, the normalization of the wave-functions are identical d 3 x 1... d 3 x N ψ E ( x P (1),..., x P (N) )ψ E ( x P (1),..., x P (N) ) = d 3 x P 1 (1)... d 3 x P 1 (N)ψ E ( x 1,..., x N )ψ E ( x 1,..., x N ) = d 3 x 1... d 3 x N ψ E ( x 1,..., x N )ψ E ( x 1,..., x N ) where, P 1 (i) is the integer that P maps onto the integer i and we have used the fact that d 3 x P 1 (1)... d 3 x P 1 (N) is an inconsequential re-ordering of d 3 x 1... d 3 x N. Then, there are two possibilities. The first possibility is that, using permutations, we have found some new quantum states which are not equivalent to the one that we began with. That is, for some permutation, P, the wave function ψ E ( x 1,..., x N ) and the wave function ψ E ( x P (1),..., x P (N) ) are truly distinct wave functions representing distinct quantum states. In order to describe distinct quantum states, the state vectors must be linearly independent. The test for linear independence is to ask whether the equation whether the equation c 1 ψ E ( x 1,..., x N )) + c 2 ψ E ( x P (1),..., x P (N) ) = 0 (11) has a solution where c 1 and c 2 are not zero. If the states are truly distinct quantum states, the only solution of equation (11) has both c 1 and c 2 equal to zero. That is, ψ E ( x 1,..., x N ) and ψ E ( x P (1),..., x P (N) ) are two different quantum states with the same energy eigenvalue E. Let us examine this possibility. If we consider all permutations and find the linearly independent states which are generated, we find a degenerate set of state vectors which are transformed into each other by permutations. The degeneracy would be a prediction of our quantum mechanical model. It us up to us to compare what we find with the real physical system which we are describing in order 10

11 to see if the degeneracies which would result are indeed there. When the degeneracy is two-fold or greater, the particles which are being described are said to obey parastatistics. In parastatistical systems, the degeneracies can depend on the total number of particles. For example, if particles are distinguishable, it could be that some or even all of the N! states that are gotten by permutations ψ E ( x P (1),..., x P (N) ) are independent and the degeneracy is N!. Nature does not seem to make use of parastatistics. 1 For any threedimensional many-particle system, and for any permutation P, the wave function ψ E ( x 1,..., x N ) and the wave function ψ E ( x P (1),..., x P (N) ) are linearly dependent and represent the same quantum state. Such particles are said to be indistinguishable. This indistinguishability is extremely important to us. It is responsible for the stability of atoms, for example, via the Pauli exclusion principle applied to identical electrons. Nature would be very different if electrons were distinguishable. When particles are indistinguishable, the equation c 1 ψ E ( x 1,..., x N ) + c 2 ψ E ( x P (1),..., x P (N) ) = 0 has a solution where both c 1 and c 2 are non-zero for any permutation P. Then, the wave-functions must be proportional to each other, ψ E ( x P (1),..., x P (N) ) = cp ] ψ E ( x 1,..., x N ) where cp ] = c 2P ] c 1. If the wave function is normalized, then cp ] = 1 and, P ] considering a permutation which, for example, exchanges the positions of just two particles, where doing the permutation twice returns the wave-function to its original form. Then c 2 P ] = 1 and we would conclude that cp ] = 1 or cp ] = 1. Then, considering the fact that any permutation can be built up out of successive interchanges of pairs of particles, we can see that, for any permutation, there are two possibilities, the first is where the wave-function is a completely symmetric function of its arguments, cp ] = 1 1 There are some examples of unusual statistics when the effective dimension of a quantum system is one or two, where permutations have a topological interpretation and the wave-function can have a richer structure. Particles which obey such statistics are called anyons. 11

12 and ψ E ( x 1,..., x N )) = ψ E ( x P (1),..., x P (N) ) for any permutation, P. The particles are called bosons, or are said to obey Bose-Einstein statistics. The second possibility is where the wave-function is a totally anti-symmetric function of the positions arguments, and cp ] = ( 1) degp ] ψ E ( x 1,..., x N ) = ( 1) degp ] ψ E ( x P (1),..., x P (N) ) for a permutation P and where the degree degp ] is the number of interchanges of pairs that are needed to implement the permutation. Particles which obey statistics of this sort are called fermions, or are said to obey Fermi-Dirac statistics. In the quantum many-body systems that are found in nature, particles that have identical properties are identical particles and they are either fermions of bosons. Given a solution of the Schrödinger equation we can construct a wavefunction for bosons by symmetrizing over the positions of the particles, so that ψ b ( x 1,..., x N, t) = c b ψ( x P (1),..., x P (N), t) P On the other hand, if the particles that the wave-function is intended to describe are fermions, then we should anti-symmetrize over the positions of the particles, ψ f ( x 1,..., x N, t) = c f ( 1) deg(p ) ψ( x P (1),..., x P (N), t) P Here, the summations are over all N! possible permutations, including the trivial one. In each of these expressions, the constants c b and c f should be adjusted to correctly normalize the resulting wave-function. When the wave-function is either completely symmetric or anti-symmetric, the probability density ρ( x 1,..., x N, t) = ψ ( x 1,..., x N, t)ψ( x 1,..., x N, t)d 3 x 1... d 3 x N is a completely symmetric function of its arguments, ( x 1,..., x N ). Since the particles are identical, the quantity ρ( x 1,..., x N, t)d 3 x 1, d 3 x 2,... d 3 x N 12

13 should be interpreted as the probability at time t for finding the system with particles occupying the infinitesimal volumes d 3 x 1, d 3 x 2,... d 3 x N which are each centered on the points x 1,..., x N, respectively, with no reference to which particles occupy which volumes. It should be normalized so that d 3 x 1... d 3 x N ψ ( x 1,..., x N, t)ψ( x 1,..., x N, t) = 1 This has the interpretation that the total probability for finding the N particles somewhere is equal to one. 2.2 Spin There is one elaboration which we should discuss before proceeding to generalize our current discussion. That is the issue of spin. If we want to describe realistic many-particle systems of atoms or electrons, the particles in question generally have spin and their wave-functions must carry an index to label the spin state. To describe these, we add an index to the total wave-function for each particle, so that the wave-function is ψ σ1 σ 2...,σ N ( x 1, x 2,..., x N, t) The indices σ i each run over J values σ i = 1,..., J which correspond to the spin states of a single particle. The wave-function of a system of identical particles must then be either symmetric or anti-symmetric under simultaneous permutations of the spin and position variables. Generally, bosons have integer spins and fermions have half-odd integer spin. In summary, for bosons, J is an integer and ψ σ1...,σ N ( x 1,..., x N, t) = ψ σp (1)...,σ P (N) ( x P (1),..., x P (N), t) For Fermions, J is a half-odd-integer and ψ σ1...,σ N ( x 1,..., x N, t) = ( 1) degp ] ψ σp (1)...,σ P (N) ( x P (1),..., x P (N), t) where, when we implement the permutation, we permute both the spin and the position labels. The Hamiltonian can also have spin-dependent interactions. In that case, the potential energy is generally a hermitian matrix which operates on spin indices. For two-body interactions, the two-body gets 13

14 spin indices as V the mapping σ iσ j ρ i ρ j ψ σ1...σ i...σ j...σ N ( x 1,..., x N, t) ( x i x j )... and its operation on the wave-function is J ρ i ρ j =1 ρ iρ j Vσ i σ j ( x i x j )ψ σ1...ρ i...ρ j...σ N ( x 1,..., x N, t) We will see shortly that this sort of interaction is very easy to implement in second quantization. 3 Second Quantization Second quantization is a technique which summarizes the many-particle quantum mechanical problem contained in (9), together with either bose or fermi statistics in an elegant way. To implement second quantization, we begin by constructing an abstract basis for the states of the N-particle system. We define the Schrödinger field operator, ψ( x) which depends on one position variable, x. In spite of the use of the symbol ψ, this operator should not be confused with a wave-function, it is an operator whose important property is that it obeys the commutation relations which will be listed in equations (12) or (13) below. There is one such operator for each different kind of identical particle, for example in a gas of electrons where the electron can exist in two spin states, the field operator would have the spin index, ψ σ ( x) with s = 1, 2 labelling the spin. We shall also need the Hermitian conjugate of the field operator, ψ σ ( x). This should be regarded as the hermitian conjugate of the operator ψ σ ( x) in the sense that (ψ σ ( x) state >) =< state ψ σ ( x), (ψ σ ( x) operator]) = operator] ψ σ ( x) In the case of particles with Bose-Einstein statistics, these operators satisfy the commutation relations ψσ ( x), ψ ρ ( y) ] = δ ρ σδ( x y), ψ σ ( x), ψ ρ ( y)] = 0, ψ σ ( x), ψ ρ ( y) ] = 0 (12) where, as usual, the square bracket denotes a commutator (A, B] = AB BA). 14

15 In the case of particles with Fermi-Dirac statistics, the commutators should be replaced by anti-commutators so that the operators satisfy the anti-commutation relations { ψσ ( x), ψ ρ ( y) } = δ ρ σδ( x y), {ψ σ ( x), ψ ρ ( y)} = 0, { ψ σ ( x), ψ ρ ( y) } = 0 (13) Here, as is conventional, we use curly brackets to denote an anti-commutator, ({A, B} = AB + BA). The operators ψ σ ( x) and ψ σ ( x) can be thought of as annihilation and creation operators for a particle at point x and in spin state σ. To see this, consider the following construction. We begin with a specific quantum state which we shall call the empty vacuum 0 >. It is the state where there are no particles at all. Its mathematical definition is that it is the state which is annihilated by the operators ψ σ ( x) for all values of the position x and spin label σ, ψ σ ( x) 0 >= 0 for all x, σ (14) The adjoint of the above statement is that the Hermitian conjugate and the dual state to the vacuum also have the property < 0 ψ σ ( x) = 0 for all x, σ (15) Then, we create particles which occupy the distinct points x 1,..., x n and are in spin states σ 1,..., σ N by repeatedly operating ψ σ i ( x i ) on the vacuum state x 1,..., x N, > σ 1...σ N = 1 ψ σ 1 ( x 1 )... ψ σ N ( x N ) 0 > (16) N! Since the operators ψ σ i ( x i ) either commute or anti-commute with each other, the state x 1,..., x N > σ 1...σ N is automatically either totally symmetric or anti-symmetric under permutations of the position coordinates and spins and it is therefore appropriate for either bosons or fermions, respectively. Similarly, The inner product is < x 1,..., x N σ1...σ N = 1 N! < 0 ψ σ1 ( x 1 )... ψ σn ( x N ) (17) < x 1,..., x N σ1...σ N y 1,..., y N, > ρ 1...ρ N 15

16 = 1 N! P ( 1) D(P ) δ( x 1 y P (1) )δ ρ P (1) σ 1... δ( x N y P (N) )δ ρ P (N) σ N where D(P ) = 0 for bosons and D(P ) = degp ] for fermions. In second quantization, the vectors x 1,..., x N, > σ 1...σ N are used to construct a state of the quantum system in the following way. A candidate for the wave-function of the system is a function of the N positions and the time, f σ1...σ N ( x 1,..., x N, t). We consider a vector in the Hilbert space of the N-particle system, f σ1...σ N ( x 1,..., x N, t) and we form the quantity f(t) >= d 3 x 1... d 3 x N f σ1...σ N ( x 1,..., x N, t) x 1,..., x N, > σ 1...σ N (18) There is a one-to-one correspondence between the state vectors f(t) > and the functions f σ1...σ N ( x 1,..., x N, t). If we have a function, f σ1...σ N ( x 1,..., x N, t), we simply form the corresponding f(t) > by forming the integrals in equation (18). If, on the other hand, we are given f(t) >, we can find the function which corresponds to it by taking the inner product, < x 1,..., x N σ1...σ N f(t) >= f σ1...σ N ( x 1,..., x N, t) (19) This gives us two languages in which we can discuss the same quantity. Now, let us assume that ψ σ1...σ N ( x 1,..., x N, t) is the wave-function. That is, it satisfies the Schödinger equation (9). Second quantization will then give us the wave-function described as the state ψ(t) >= d 3 x 1... d 3 x N ψ σ1...σ N ( x 1,..., x N, t) x 1,..., x N, > σ 1...σ N (20) Unit normalization of the wave-function ψ σ1...σ N ( x 1,..., x N, t) results in unit normalization of the state Ψ(t) >, < Ψ(t) Ψ(t) >= d 3 x 1... d x N 2 σ 1...σ N =1 = 1 ψ σ 1...σ N ( x 1,..., x N, t)ψ σ1...σ N ( x 1,..., x N, t) We can ask the question as to what is the equation which Ψ(t) > must satisfy that is equivalent to the fact that ψ σ1...σ N ( x 1,..., x N, t) satisfies the 16

17 Schrödinger equation. To answer this question, we consider the operator H = d 3 x 2 σ=1 d xd 3 y h 2 2m ψ σ ( x) ψ σ ( x) 2 ψ σ ( x)ψ ρ ( y)v ( x y)ψ ρ ( y)ψ σ ( x) (21) σ,ρ=1 This operator is the Hamiltonian in the second quantized language. It is easy to see that ψ σ1...σ N ( x 1,..., x N, t) obeys the Schrödinger equation (9) when Ψ(t) > satisfies the equation i h Ψ(t) >= H Ψ(t) > (22) t Furthermore, one can construct an initial state Ψ(0) > using the initial many-particle wave-function ψ σ1...σ N ( x 1,..., x N, t = 0). The state at later times is then uniquely determined by (22) which has the formal solution Ψ(t) >= e iht/ h Ψ(0) > and the wave-function at any time that can be extracted from it by taking the inner product, ψ σ1...σ N ( x 1,..., x N, t) =< x 1,..., x N σ1...σ N Ψ(t) > and it must coincide with the solution of the many-body Schrödinger equation (9). Thus, the mathematical problem of solving the second-quantized operator equation (22) is identical in all respects to the mathematical problem of solving the many-particle Schrödinger equation (9), they are solved when we find the wave-function ψ σ1...σ N ( x 1,..., x N, t) or equivalently the state Ψ(t) >. We thus have two equivalent formulations of the same theory. The state x 1... x N > σ 1...σ N that we have constructed should be thought of as the quantum mechanical state where the N particles can be found occupying the positions x 1,..., x N. To see this, we note that x 1... x N > σ 1...σ N is an eigenstate of the density operator, which we form from the product of a creation and annihilation operator, ρ( x) = 2 ψ σ ( x)ψ σ ( x) σ=1 17

18 Operating on x 1,..., x N > σ 1...σ N, we discover that these states are eigenstates of the density with eigenvalues given by a sum of delta-functions ( N ) ρ( x) x 1... x n > σ 1...σ N = δ( x x i ) x 1... x n > σ 1...σ N (23) i=1 which is just what we would expect for a group of N particles localized at positions x 1,..., x N. This formula holds for both bosons and fermions. The second quantized Schrödinger equation (22) does not contain the explicit information that there are N particles. The number of particles can be measured by the number operator which is an integral over space of the density operator, 2 N = d 3 x ψ σ ( x)ψ σ ( x) σ=1 The states, Ψ(t) >, which we have constructed are eigenstates of the particle number operator, N Ψ(t) > = N Ψ(t) > Furthermore, the Hamiltonian commutes with the number operator, N, H] = 0 (24) This can easily be checked explicitly using the algebra for the operators ψ( x) and ψ ( x). The result is that the number operator N and the Hamiltonian H can have simultaneous eigenvalues and that the total number of particles will be preserved by the time evolution of the system. 4 The Heisenberg picture Let us pause to review what we have done so far. We have found two different descriptions of the many-particle quantum system. The first one is conventional quantum mechanics where we should solve the partial differential equation to find the wave-function of the many-particle system. We have summarized the relevant equations in Infobox 4.1 below. 18

19 Infobox 4.1 Many-particle quantum mechanics Many-particle Schrödinger equation: i h t ψ( x N h 2 2 1,..., x N, t) = i 2m + i=1 i<j V ( x i x j ) ] ψ( x 1,..., x N, t) Normalization of wave-function: d 3 x 1... d 3 x N ψ ( x 1,..., x N, t)ψ( x 1,..., x N, t) = 1 Bosons: Fermions ψ( x 1,..., x N, t) = ψ( x P (1),..., x P (N), t) ψ( x 1,..., x N, t) = ( 1) deg(p) ψ( x P (1),..., x P (N), t) for any permutation P. If the particles have spin, the wave-functions has spin indices: ψ( x 1,..., x N, t) ψ σ1...σ N ( x 1,..., x N, t) If the interactions are spin-dependent, the two-body potential becomes a hermitian matrix operating on spins, V ( x i x j )ψ( x 1,..., x N, t) i<j i<j J ρ i ρ j =1 ρ iρ j Vσ i σ j ( x i x j )ψ σ1...ρ i...ρ j...σ N ( x 1,..., x N, t) The other is the second quantized picture, where we are given Hamiltonian and number operators containing fields and the operator nature of the fields is defined by their commutation relations. These and the equation of motion for the quantum state are summarized in Infobox 4.2 below. We have seen in the above development how the mathematical problem which is defined in 19

20 Infobox 4.1 and Infobox 4.2 are equivalent. Infobox 4.2 Second quantization in the Schrödinger Picture Schrödinger equation: i h Ψ(t) >= H Ψ(t) >, N Ψ(t) >= N Ψ(t) > t Number and Hamiltonian operators: N = H = d 3 x d 3 x 2 ψ σ ( x)ψ σ ( x) σ=1 2 h 2 ψ 2m σ ( x) ψ σ ( x) σ= d xd 3 y J ψ σ ( x)ψ σ ( y)v σρ=1 ρρ σσ ( x y)ψ ρ ( y)ψ ρ ( x) For generality, we have included possible spin-dependence of the twobody potential. Commutation relations: ψσ ( x), ψ ρ ( y) ] = δρ σδ( x y) ψ σ ( x), ψ ρ ( y)] = 0, ψ σ ( x), ψ ρ ( y) ] = 0 For compactness of notation, we use A, B] = AB BA for commutator and anti-commutator. wherea, B] = AB BA is the commutator for bosons and the anticommutator for fermions. The latter set of equations, those in Infobox 4.2, are essentially the definition of a quantum field theory. The quantum fields are the field operators ψ σ ( x) and ψ σ ( x). Note that they do not depend on time. Instead, the state Ψ(t) > is time dependent. The reason for this is 20

21 that, like our many-particle problem (9), we have formulated the problem in the Schrödinger picture of quantum mechanics where operators are time independent and the states carry the time dependence. The Heisenberg picture is an alternative and equivalent formulation of quantum mechanics. It is related to the Schrödingier picture that we have developed so far by a time dependent unitary transformation of the operators and state vectors. The unitary transformation begins with the observation that, if we know the state of the system at an initial time, say at t = 0, we can find a formal solution of the equation of motion for the state vector, Ψ(t) = e iht/ h Ψ(0) which uses a unitary operator that is obtained by exponentiating the Hamiltonian, exp ( iht/ h). We can thus set the state vector to its initial condition (assuming t = 0 is where we must impose an initial condition) by a unitary transformation. Going to the Heisenberg picture simply does this unitary transformation to all of the operators to get an equivalent description of the theory where the operators are time-dependent and the states are independent of time. The unitary transformation of the operators is ψ σ( x, t) = e iht/ h ψ σ ( x)e iht/ h, ψ σ ( x, t) = e iht/ h ψ σ ( x)e iht/ h (25) In the Heisenberg picture, the states are time independent. For a given physical situation the quantum state is simply given by the initial state of the system. The operators, on the other hand, become time dependent and it is their time dependence which carries the information of the time evolution of the quantum system. In quantum field theory, particularly the relativistic quantum field theory which we shall study later on, the equations of motion are more commonly presented in the Heisenberg picture. Unlike the time-independent operators ψ σ ( x) and ψ σ ( x) which we introduced in order to construct second quantization, the Heisenberg picture operators ψ σ( x, t) and ψ σ ( x, t) now depend on time and their time dependence contains dynamical information, which is determined by equations (25). This information can also be given as a differential equation, the Heisenberg equation of motion, which can be obtained by taking a time derivative of equations (25). i h t ψ( x, t) = ψ( x, t), H], i h t ψ ( x, t) = ψ ( x, t), H ] (26) 21

22 These are the usual algebraic operator equations which are meant to be solved to find the time dependence of the operators in the Heisenberg picture. In (26), and elsewhere when it is clear from the context, we will drop the prime on the Heisenberg picture fields. They are still distinguished from the Schrödinger picture fields in that they are time dependent and the Schrödinger picture fields are not. The Heisenberg picture field operators have an equal-time commutator algebra, which can be obtained from (12) or (13) by multiplying from the left and right by e iht/ h and e iht/ h, respectively. This leads to the canonical equal-time commutation relations for Bosons, ψσ ( x, t), ψ ρ ( y, t) ] = δσδ( x ρ y) (27) ψ σ ( x, t), ψ ρ ( y, t)] = 0, ψ σ ( x, t), ψ ρ ( y, t) ] = 0 (28) or the canonical equal-time anti-commutation relations for Fermions, { ψσ ( x, t), ψ ρ ( y, t) } = δ ρ σδ( x y), (29) {ψ σ ( x, t), ψ ρ ( y, t)} = 0, { ψ σ ( x, t), ψ ρ ( y, t) } = 0 (30) At this point the reader should take careful note of the fact that this algebra holds only when the times in the operators are the same. The time-derivative of the time-dependent field ψ σ ( x, t) can be computed from the Heisenberg equation of motion (26) using the equal time commutation relations. It is given by an equation which looks like a non-linear generalization of the Schrödinger equation ) h2 (i h + t 2m 2 ψ σ ( x, t) = d 3 yv ρρ σσ ( x y )ψ σ ( y, t)ψ ρ ( y, t)ψ ρ ( x, t) (31) Again, the nonrelativistic quantum mechanics problem is presented as a quantum field theory. The operators ψ σ ( x, t) and ψ σ ( x, t) are the quantized fields. They satisfy the equal time commutation relations in (27) and (28) or (29) and (30). These define their algebraic properties as quantum mechanical operators. Their time evolution is determined by solving the non-linear field equation (31). That field equation has been presented in a standard form, with the Schrödinger wave operator i h t + h2 2m 2 22

23 operating on the field and with an additional non-linear interaction term. We should note the similarity of the field equation with the non-linear Schrödinger equation for a single particle. However, as we have said before, ψ σ ( x, t) is not a wave-function of a single particle, it is an operator which obeys the equal time (anti-)commutation relations. We thus have our third presentation of the many-particle problem, the field equation of a quantum field theory plus the equal-time commutation or anti-commutation relations which define the quantum fields as operators, Infobox 4.3 Second quantization in the Heisenberg picture Field equation: ) h2 (i h + t 2m 2 ψ σ ( x, t) = d 3 yv ρρ σσ ( x y)ψ σ ( y, t)ψ ρ ( y, t)ψ ρ ( x, t) Equal-time commutation relations: ψσ ( x, t), ψ ρ ( y, t) ] = δρ σδ( x y), ψ σ ( x, t), ψ ρ ( y, t)] = 0, ψ σ ( x, t), ψ ρ ( y, t) ] = 0 A spin-independent interaction has the form V ρρ σσ ( x y) = δσδ ρ ρ σ v( x y) The number of particles could still be enforced by constraining the states to be those which are eigenstates of the number operator with eigenvalue N, the number of particles, N phys >= N phys >, N = d 3 x ψ σ ( x, t)ψ σ ( x, t) where the latter should be solved by initial data, an initial state Ψ > which has N particles N Ψ >= N Ψ >. We note here that, since the number operator N commutes with the Hamiltonian, it is independent of the time. 23

24 References 1] A. A. Abrikosov, L. P. Gorkov, L. E. Dzialoshinski Methods of Quantum Field Theory in Statistical Physics, Dover Books on Physics, ] A. L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems, Dover Books on Physics, ] Richard D. Mattuck, A Guide to Feynman Diagrams in the Many-Body Problem, Dover Books on Physics,