In quantum mechanics, the number of particles is conserved. Yet,  QM can be realized in the Fock space: in this case each N-particle sector is invariant with respect to the Hamiltonian, time evolution, and other inertial transformations from the Poincare group. This is a pretty accurate representation of reality. The only missing component is the possibility of particle creation and destruction, such as emission and absorption of photons. 

The role of QFT is to fill this gap and add to the QM Hamiltonian a missing perturbation responsible for the particle creation/destruction processes.  So, there should be no fundamental differences between the two theories.  The QFT Hamiltonian (expressed in terms of physical particles)  must be equal to the familiar QM  Hamiltonian plus small terms describing the crossing of boundaries between Fock space sectors.

 Unfortunately, the QFT Hamiltonian is not presented in this form in textbooks. This is because QFT is usually formulated in terms of bare particles, which have no relevance to the particles observed in nature.  For a more physical formulation of QFT, check the "dressed particle" or "clothed particle" theory. 

If you have a quantum state | n > in the Dirac notation, and Psi(x) is a quantum field, you may define a wave function

psin(x) = < O | Psi(x) | n >,

where

Luiz@

As you say one must use imaginary time to give rigorous meaning to the path integral. But isn't it sufficient to have just-a-little amount of imaginary time. I.e., can't you let  

t -> t - i eps --- or perhaps t -> t (1-i eps),

with t real and eps real and positive, to give a proof of its existence?

My answer is that there is absolutely no relationship between wave functions and quantum fields.

Wave functions are probability amplitudes that can be squared and integrated in order to get probabilities of measurements.

Quantum fields are abstract operator functions, linear combinations of particle creation and annihilation operators, which are good only as "building blocks" for interaction operators in QFT.  The best book explaining this point of view is Weinberg's "The quantum theory of fields" vol. 1.

It is very unfortunate that people often use the same letter \psi for both wave functions and quantum fields. This makes a wrong  impression of some connection between the two.

f(q) is wave function, being an element of the Hilbert space, and also a "field" in QFT.

Edward,

I guess you are distinguishing between a `mathematical description'

and the `reality that is mathematically described'. The `wave

function' is part of a mathematical description. This term is

usually used in case it is describing one or a few particles. More

generally, the term `state function' or `state vector' is used.

If the object is a quantum field (containing many microscopic

particles) the latter terms are usually used. Since they are just

mathematical descriptions, state vectors are even defined in case

the field does not contain a well-defined number of particles (for

instance, the coherent states of light).

Hence, the `relation between the wave function in quantum mechanics

and a field in quantum field theory' is one of analogy and

generalization: analogy between different, in QFT even generalized,

objects (viz. `a few particles' and a `field containing a possibly

uncertain number of particles').

Since in both cases a state vector can be mathematically defined as

a vector in some vector state, the vector's evolution in time can

be mathematically described as a path in that vector space. This

should in general be distinguished from the path of a single

particle in physical space (which is just a very special case in

which the mathematical representation of the space is so obvious

that we usually forget about the difference between theory and

reality). In quantum theory you better never forget about the difference between theory and reality.

Dear everybody,

before getting into all sort of subtleties, let's make clear that the wave-function is for the 1st quantization, and the field operators are for the 2nd quantization. There is not a relation, between the two, but a difference.

The 1st quantization works with a fixed number and type of particles. In the 2nd quantization particles may transform from one type to another, and also we may have to do with states in which the number of particles is simply not fixed. A good illustration of the latter is the state describing a laser beam - the so-called coherent state.

So, in the 2nd quantization we have field operators containing creation and annihilation operators. There are no such "creatures" in the 1st quantization.

Thanks everyone for taking the time to answer this, I certainly appreciate it.  I had read Weinberg's quantum theory of fields where he also argues the wave function / state vector in QM is quite distinct from a field in QFT (he doesn't even like the term second quantization).  But have also read McMahon's quantum field theory demystified where the second quantization is described as quantizing the wave function in the Dirac equation (in the case of the electron field), and he seems to be arguing this amounts to promoting the wave function to a field in QFT.  What are all of your thoughts on this?

Willem, your response was certainly the closest in spirit to what I was wondering about.   Is a wave functional in QFT the mathematical entity describing the state of a field (analogous to wave function describing state of particle)?

Sofia and Luiz, are Fock states in QFT energy eigenstates?  And does this depend on if particles in field are interacting or not?

thanks again all!

In quantum mechanics, the number of particles is conserved. Yet,  QM can be realized in the Fock space: in this case each N-particle sector is invariant with respect to the Hamiltonian, time evolution, and other inertial transformations from the Poincare group. This is a pretty accurate representation of reality. The only missing component is the possibility of particle creation and destruction, such as emission and absorption of photons. 

The role of QFT is to fill this gap and add to the QM Hamiltonian a missing perturbation responsible for the particle creation/destruction processes.  So, there should be no fundamental differences between the two theories.  The QFT Hamiltonian (expressed in terms of physical particles)  must be equal to the familiar QM  Hamiltonian plus small terms describing the crossing of boundaries between Fock space sectors.

 Unfortunately, the QFT Hamiltonian is not presented in this form in textbooks. This is because QFT is usually formulated in terms of bare particles, which have no relevance to the particles observed in nature.  For a more physical formulation of QFT, check the "dressed particle" or "clothed particle" theory. 

Edward,

De distinction between functions and functionals is a mathematical

one, and is not related to the distinction between particles and

fields. For instance, functionals are applied in the quantum

mechanics of one particle in case a position representation is

used. The eigenstates of the position operator are functionals

(although referred to as Dirac delta-functions, however, this has

later been made mathematically rigorous by Schwartz by developing a

theory of functionals or distributions). Since fields are

infinite-dimensional their states usually behave in an analogous

singular way as delta functions, and hence should be mathematically

described by functionals. However, fields may just as well have

perfectly regular states as particles may have. Coherent states are

examples of this.

Quantum field theory describes many particle systems. We have to therefore compare QFT with quantum mechanics of a many particle system. Non-relativistic QM does not include the notion of particle creation and destruction, but QFT does include this aspect. Furthermore in QFT particles are identical but in QM of a many particle system particles are usually taken to be distinct.  For a useful description of identical particles one has to move from Hilbert space to Fock space. Fock space is spanned by the following states. 1) There is no particle, which is well known vacuum state  2) There is one particle 3) There are two particle ..... there are N particles. The general state of a many particle system is (therefore) a linear combination of these N basis states. At this point the quantum mechanical many particle state becomes a quantum field. Degrees of freedom of a quantum field being occupation numbers. Basis states with different occupation numbers are connected with each other by creation and annihilation operators.

Fock space can also be described as a symmetric or anti-symmetric tensor product space of N single particle Hilbert spaces.

Ref:

A Fock space is aHilber

To understand what is mathematically going on when a 'field', which is specified by some field amplitudes f_1(x),...,f_n(x) and a Lagrangian density, is 'quantized' it is useful to study the mathematical rigorous way how the e.m. field (A_0,A_1,A_2,A_3) is 'quantized'. This is done by starting from a Hilbert space H of functions and by constructing tensor products of H for each number of 'field lumps' (i.e. particles) and finally 'adding' them to obtain the Fock space F(H). In F(H) the machinery of creation operators and annihilation operators can be defined 'rigorously'. 'Rigorously' means that wave packets are considered instead of plane waves. The field amplitudes are associated with field 'operators', or more precisely 'operator-valued' distributions, which act in Fock space. An element of Fock space describes a system consisting out of 'independent' lumps of e.m. energy, which can be defined as 'photons'. Proceeding in the same way, but using anticommutators instead of commutators for creation ops. and annihil. ops, the Schroedinger field (considered as a 'classical' matter field) leads to the formalism of non.rel. many-particle-QM (without spin), and the Dirac field in conjunction with the e.m. field leads to QED.

All this stuff are mathematical formalisms. Physics enters when a physical meaning is associated

(1) with the field amplitudes

and (2) with the currents corresponding to symmetries of the Lagrangian density (Noether theorem)

and (3) with the scalar product of Fock space vectors |A> and |B>.

REMARK concering (3):

Usually is considered as a "probability amplitude" that an object characterized by Fock space vector |A> can be (most generally speaking) considered to "some part" as an object characterized by Fock space vector |B>, and by hypothesis "some part" (the "probability") is given by ||**2, i.e. the "probability credo" of QM and QFT.

REMARK concering (2):

By means of the machinery of the Lagrange formalism for fields a tensor for energy-momentum density and tensor for angluar-momentum density and tensors for charge densities emerge.

REMARK concering (1):

(A) In the case of the e.m. field the field amplitudes A_mue is the 4-vector potential which is related to the el. and magn. field strength E and B in the usual manner.

(B) In the case of the Schroedinger field it can be shown that the "wave function" PSI(x) of ordinary QM is equal to where "vac" is the so-called vacuum vector in Fock space and psi(x) is the field operator of the Schroedinger field acting in Fock space, and |PSI> is a single particle vector in Fock space. In the case of n particles the wave function is PSI(x_1,...x_n) = where |PSI> is a n-particle vector in Fock space. Although this field-theoretic set-up of QM is mathematically a little bit  sophisticated, it can be considered as a particularly convincing approach to non.rel. QM for describing electrons interacting with atoms or molecules or other material objects.

(C) In the case of the Dirac field in conjunction with e.m. field and in the case of other fields with some non-abelian gauge symmetry or pure gauge fields it seems to be best (up to now) to consider the field amplitudes just as a means to form "wave packets", i.e. tiny lumps having energy and momentum and charge according to the energy-momentum density and charge densities of the field which is specified by a Langrangian density.