What Are the States in Quantum Field Theory?

[Azrael84]

Hello,

this is quite a basic question I know, but something I'm not sure I've fully got my head around. In classical particle mechanics the dynamical variable is the position vector x, and in classical field theory the dynamical variable becomes the field $$\phi(x)$$, with x being relagated to just a label to each of the infinitity of these dynamical variables $$\phi$$ at each point in space if you like.

Then in regualar QM, the things that were dynamical variables in classical particle mechanics (position, momentum etc...) get promoted to operators, with the actual states being abstract vectors in the hilbert space of these operators. In QFT the dynamical variable from classical field theory, the field itself, gets promoted to an operator. But what are the analogues of the eigenvalues of position/momentum/angular momentum etc from regualar QM?

Also I noticed that in QM the solutions of the Schrodinger equation are the actual states themselves, but in QFT the solutions of the KG eqn or the Dirac equation say, are the field operators not the states. So what are the actual states? are they somehow abstract vectors of the field operators?

 

[meopemuk]

QFT is not fundamentally different from quantum mechanics. In QFT we also have a Hilbert space (it is called the Fock space) and states of the system are represented by vectors in this Hilbert space. The most significant difference is that the number of particles is not fixed in QFT. So the Fock space is a direct sum of subspaces with 0 particles (vacuum), 1 particle, 2 particles, etc. Thus in the Fock space we can describe processes in which the number of particles can change (radiation, decays, etc.). So, basically QFT is the same as "QM with variable number of particles".

Quantum fields have nothing to do with wave functions or states. Quantum fields are just certain operators in the Fock space which are convenient "building blocks" for construction of other more physical operators. For example, interaction terms in the Hamiltonian are usually constructed as products of several field operators.

Eugene.

 

[Fredrik]

The Hilbert space of one-particle states of a QFT without interactions can be constructed explicitly by taking the vectors to be equivalence classes of square integrable positive-frequency solutions of the classical field equation. But it doesn't really matter what the Hilbert space is, since all separable infinite-dimensional Hilbert spaces are isomorphic to each other. I suppose the point of the explicit construction is that it makes it easy to explicitly construct the generators of (an irreducible representation of) the Poincaré group (i.e. the (four-)momentum, angular momentum and boost operators) from the quantum field.

The Hilbert space of many-particle states is constructed as the Fock space of the Hilbert space of one-particle states.

When there are interactions in the theory, things get really complicated, and I don't understand this well enough to comment. (The only person here who really seems to understand these things is DarMM).

 

[meopemuk]

When there are interactions in the theory, things get really complicated...

You are right, QFT gets really complicated (physical particles are not the same as bare particles; the need for renormalization, etc) if interaction in the Hamiltonian has terms which create additional particles in 0-particle and 1-particle states. In the creation/annihilation operator notation these troublesome interaction terms may look like a*a*a or a*a*a*.

There is no good reason why such bad terms should be present in realistic Hamiltonians. One can build successfult QFT theories without these bad interactions. For example, there is a "dressed particle" version of QED, in which the simplest interaction term has the form a*a*aa. In this theory particles cannot appear spontaneously out of vacuum or 1-particle states. There is no difference between physical particles and bare particles. There is no need for renormalization. The entire theory is not more complicated than standard quantum mechanics.

Eugene.

 

[samalkhaiat]

Also I noticed that in QM the solutions of the Schrodinger equation are the actual states themselves,

This is also true in QFT. They are called wavefunctional, $\Psi[\psi] = <\psi|\Psi(t)>$ and the Schrodinger equation of QFT is the following functional differential equation

$$i\partial_{t}\Psi[\psi] = \int d^{3}x \mathcal{H}(\psi , \frac{\delta}{\delta \psi(x)})\Psi[\psi]$$

Solving this equation can reproduce all QFT results. However, that is a very hard job.

QFT the solutions of the KG eqn or the Dirac equation say, are the field operators not the states.

There is a deep mathematical reason for that. The KG "field" and Dirac spinor can not be regarded as wavefunctions in Hilbert space but fields on Minkowski spacetime. The reason for this is the NON-COMPACT nature of the Poincare' group. All FINITE-DIMENSIONAL
irreducible representations of any non-compact group are NOT UNITARY. Therefore these IR representations can not be carried by functions on Hilbert space, but by functions on spacetime where non-unitarty does not cause any problem.


regards

sam

 

[meopemuk]

samalkhaiat,

From what you've said I guess that the wavefunctional $\Psi[\psi] = <\psi|\Psi(t)>$ cannot be considered as a probability amplitude. Quantum field $\Psi(t)$ transfrorms by a non-unitary representation of the Poincare group. Therefore the total probability (which must be equal 1 in all reference frames) is not necessarily preserved in this representation. Is that correct?

Eugene.

 

[hamster143]

But it doesn't really matter what the Hilbert space is, since all separable infinite-dimensional Hilbert spaces are isomorphic to each other.

Isomorphic does not mean homeomorphic. Just like R is isomorphic to R^2, but not homeomorphic to it. Hilbert space isomorphisms are useless if they are not homeomorphisms.

 

[hamster143]

From what you've said I guess that the wavefunctional $\Psi[\psi] = <\psi|\Psi(t)>$ cannot be considered as a probability amplitude. Quantum field $\Psi(t)$ transfrorms by a non-unitary representation of the Poincare group. Therefore the total probability (which must be equal 1 in all reference frames) is not necessarily preserved in this representation. Is that correct?

$\Psi[\psi(x)] = <\psi(x)|\Psi(x,t)>$ denotes an amplitude to find the wavefunctional in the state $\psi(x)$ everywhere on a hypersurface t=const. It is inherently Lorentz non-invariant. Since the hypersurface is not invariant under general Poincare transform, I'm not sure if the concept of representation even applies here.

If you work with a subset of Poincare group that preserves the hypersurface (it consists of translations and SO(3) rotations), the probability is certainly conserved

 

[Azrael84]

This is also true in QFT. They are called wavefunctional, $\Psi[\psi] = <\psi|\Psi(t)>$ and the Schrodinger equation of QFT is the following functional differential equation

$$i\partial_{t}\Psi[\psi] = \int d^{3}x \mathcal{H}(\psi , \frac{\delta}{\delta \psi(x)})\Psi[\psi]$$

Solving this equation can reproduce all QFT results. However, that is a very hard job.

So how do these wavefunctionals compare to the usual $$\mid 0\rangle$$, $$\mid k_1\rangle$$ etc etc, type of states that one normally sees when learning QFT? are they the same thing? (is it just like in QM, where one has the wavefunction as the position representation of the abstract vector, here the wavefunctional would be the "$$\phi$$ representation"?).

Why do these wavefunctionals obey the Schroedinger equation too? and not a Lorentz invariant equation like KG? I don't understand how the non-relativistic Schroedinger can have a role in QFT.

 

[Fredrik]

Isomorphic does not mean homeomorphic. Just like R is isomorphic to R^2, but not homeomorphic to it. Hilbert space isomorphisms are useless if they are not homeomorphisms.

It does when we're talking about Hilbert space isomorphisms. They preserve both the vector space structure and the inner product. The latter requirement implies that they're bounded, and that implies that they must be continuous. Their inverses must be continuous too. So Hilbert space isomorphisms are homeomorphisms.

R and R^2 is a strange example, since they clearly aren't vector space isomorphic. You must have meant that they're isomorphic sets, i.e. that there's a bijection from one of the sets into the other, but that's not the kind of "isomorphism" we're interested in.

 

[Fredrik]

I don't understand how the non-relativistic Schroedinger can have a role in QFT.

There must exist a time translation operator which satisfies $U(t_1+t_2)=U(t_1)U(t_2)$ and this gives us $$U(t)=e^{-iHt}$$. This last equation is the definition of the Hamiltonian in both special relativistic and non-relativistic QM. States satisfy the Schrödinger equation because the time evolution operator does:

$$i\frac{d}{dt}|\psi(t)\rangle=i\frac{d}{dt}\Big(e^{-iHt}|\psi\rangle\Big)=He^{-iHt}|\psi\rangle=H|\psi(t)\rangle$$

The difference between special relativistic and non-relativistic QM is that we have to consider representations of the Poincaré group instead of the Galilei group, but the time translation group is a subgroup of both.

 

[Azrael84]

There must exist a time translation operator which satisfies $U(t_1+t_2)=U(t_1)U(t_2)$ and this gives us $$U(t)=e^{-iHt}$$. This last equation is the definition of the Hamiltonian in both special relativistic and non-relativistic QM. States satisfy the Schrödinger equation because the time evolution operator does:

$$i\frac{d}{dt}|\psi(t)\rangle=i\frac{d}{dt}\Big(e^{-iHt}|\psi\rangle\Big)=He^{-iHt}|\psi\rangle=H|\psi(t)\rangle$$

The difference between special relativistic and non-relativistic QM is that we have to consider representations of the Poincaré group instead of the Galilei group, but the time translation group is a subgroup of both.

Thanks.

 

[Azrael84]

I was already aware of the Fock space being the analogue of the Hilbert space in QFT when I originally posted. I guess what I was really wondering is what are the states in the "non-abstract sense"? e.g. in QM you could choose to work in the position rep, express the operators of your Hilbert space in their position rep, and have wavefunctions that are dependent on space. So what I'm wondering is can you do something equivalent in QFT...after so called "second quantization", the operators P, X from QM have been relagated to labels, and the things satisfying the Schroedinger equation (or the KG or Dirac equation), that were our states in QM (Wavefunctions or more abstract Hilbert vectors) have been promoted to operators. But the new states of these field operators, are at the moment just abstract Fock vectors in my mind at the moment, I can't quite see how to visualize them, and am still left with the question what actually are they?

They are not configurations of the field, since the field is the operator, but then what are they...

Perhaps then they are these "wavefunctionals", things are essentially functions of the field config which is itself a function of spacetime. But I'm having difficultyl reconciling that view with the fock kets notion.

 

[meopemuk]

If you work with a subset of Poincare group that preserves the hypersurface (it consists of translations and SO(3) rotations), the probability is certainly conserved

Since boosts do not preserve "hypersurfaces", does that mean that for a moving observer the total probability of events does not necessarily add up to 1? Does that mean that probability is not longer a physically relevant quantity in QFT?

Eugene.

 

[meopemuk]

I was already aware of the Fock space being the analogue of the Hilbert space in QFT when I originally posted. I guess what I was really wondering is what are the states in the "non-abstract sense"? e.g. in QM you could choose to work in the position rep, express the operators of your Hilbert space in their position rep, and have wavefunctions that are dependent on space. So what I'm wondering is can you do something equivalent in QFT...after so called "second quantization", the operators P, X from QM have been relagated to labels, and the things satisfying the Schroedinger equation (or the KG or Dirac equation), that were our states in QM (Wavefunctions or more abstract Hilbert vectors) have been promoted to operators. But the new states of these field operators, are at the moment just abstract Fock vectors in my mind at the moment, I can't quite see how to visualize them, and am still left with the question what actually are they?

They are not configurations of the field, since the field is the operator, but then what are they...

Perhaps then they are these "wavefunctionals", things are essentially functions of the field config which is itself a function of spacetime. But I'm having difficultyl reconciling that view with the fock kets notion.

Azrael84,

You can visualize QFT states and wavefunctions just as easily as you can do that in usual quantum mechanics.

The Fock space of QFT is a direct sum of subspaces (or sectors) with different number of particles: 0, 1, 2, 3,... Take for example the 2-particle sector. In this sector you can define operators of observables (position, momentum, spin, etc.) of both particles exactly as you do that in any 2-particle problem in QM. You can also define eigenvectors/eigenvalues of these operators. So, you can form corresponding orthonormal bases and define wavefunctions in any convenient representation (position, momentum, etc.).

The same things can be done in any N-particle sector of the Fock space. This sector looks the same as the Hilbert space of N-particle quantum mechanics. The only important difference between QFT QM is that in the QFT Fock space you can now consider states which do not have a definite number of particles. Such states can have non-zero projections on sectors with different N. Wave functions of such a states can be written as collections of N-particle wave functions with different N.

Eugene.

 

[hamster143]

Since boosts do not preserve "hypersurfaces", does that mean that for a moving observer the total probability of events does not necessarily add up to 1? Does that mean that probability is not longer a physically relevant quantity in QFT?

Eugene.

That means that states do not transform in a trivial way.

In translations, the transformation law is simply $\Psi'[\psi(x)] = \Psi[\psi(x+\Delta)]$. (Since psi is an arbitrary function, what we have here is an infinite-dimensional representation of the symmetry group. Most people would be scared at this point, but we'll move on.)

In rotations, we have to consider the possibility that psi has spin and different components go into one another under rotation. That's still fairly easy.

To compute the outcome after a boost, we have to compute the wavefunction on the "new" hypersurface from the wavefunction on the "old" hypersurface. To do that, we need dynamics of the field and the exact Hamiltonian.

To correct my earlier remark, I think that the transformation law is still a unitary representation, but it's not like any usual representations we normally see.

 

[Azrael84]

The Fock space of QFT is a direct sum of subspaces (or sectors) with different number of particles: 0, 1, 2, 3,... Take for example the 2-particle sector. In this sector you can define operators of observables (position, momentum, spin, etc.) of both particles exactly as you do that in any 2-particle problem in QM. You can also define eigenvectors/eigenvalues of these operators. So, you can form corresponding orthonormal bases and define wavefunctions in any convenient representation (position, momentum, etc.).

Thanks. You say "eingenvectors/eigenvalues of these operators" but what operators? the usual momentum/position operators of the fixed dimensional Hilbert space of QM, have been turned into labels. The only operators I'm aware of the field itself $$\phi$$, and I have no idea what eigenvalues eigenvectors of this actually are, or how to view them in a particular representation.
How do you define the momentum operator in say the 3-particle sector of the Fock space?

 

[meopemuk]

How do you define the momentum operator in say the 3-particle sector of the Fock space?

Let us start with the 1-particle sector. This subspace carries a unitary irreducible representation of the Poincare group (see Weinberg, vol. 1 chapter 2). This representation allows us to define all relevant 1-particle operators (momentum, spin, position, mass, etc.) there.

The 3-particle sector is a (appropriately symmetrized or antisymmetrized) tensor product of 3 one-particle Hilbert spaces. We've already build 1-particle observables in each 1-particle Hilbert space. These observables transfer to the 3-particle sector according to the mapping associated with the tensor product construction. So, there is no difficulty in defining 1-particle momentum operators p1, p2, and p3. These operators commute with each other. So, one can find their common eigenvectors and define wave functions in the momentum representation. This is not different from ordinary quantum mechanics.

Eugene.

 

[Azrael84]

Let us start with the 1-particle sector. This subspace carries a unitary irreducible representation of the Poincare group (see Weinberg, vol. 1 chapter 2). This representation allows us to define all relevant 1-particle operators (momentum, spin, position, mass, etc.) there.

I don't have access to Weinberg right now, but I will check it out when I get chance. For now could you tell me then what the explicit form of the, say momentum, operator is then in the 1-particle sector, in say, position rep? Because it surely isn't just $$\hat{P}=-i\partial_x$$ like in ordinary QM position rep. Also for that matter what is the explicit form of the state $$\mid p \rangle$$ in the position rep, in the 1-particle sector.

 

[meopemuk]

For now could you tell me then what the explicit form of the, say momentum, operator is then in the 1-particle sector, in say, position rep? Because it surely isn't just $$\hat{P}=-i\partial_x$$ like in ordinary QM position rep.

Why not? The momentum operator will be exactly as you wrote it.

Also for that matter what is the explicit form of the state $$\mid p \rangle$$ in the position rep, in the 1-particle sector.

It will be the usual plane wave.

For more details you might find useful chapter 5 in http://www.arxiv.org/abs/physics/0504062

Eugene.

 

[Fredrik]

"Special topics in particle physics", by Robert Geroch (unpublished) has a lot to say about the explicit construction of the one-particle space and some operators on it. It was a bit hard to find today since the main link to the article has stopped working, but I was able to find one that works by doing a google search for "geroch special topics filetype:pdf". If the link below doesn't work when you read this, try that search.

http://strangebeautiful.com/other-texts/geroch-qft-lectures.pdf

 

[LAHLH]

I am new to QFT, so forgive me if anything I say is incorrect, but most of this is based on what I read in David Tong's excellent QFT notes available online.

Starting with purely classical field theory, we can use Noether's theorem to find the classical expression for the total value of a field. Noether's theorem says that if we have a symmetry, a symmetry being a change in the field $$\delta\phi_a(x)=X_a(\phi)$$, such that $$\delta\mathcal{L}=\partial_{\mu}F^{\mu}$$. Then this gives rise to a conserved current $$j^{\mu}=\tfrac{\partial\mathal{L}}{\partial(\partial_{\mu}\phi_a)}X_a(\phi)-F^{\mu}(\phi)$$, satisfying $$\partial_{\mu}j^{\mu}=0$$. Now consider the space-time translation invariance, $$x^{\nu}\rightarrow x^{\nu}-\epsilon^{\nu}\Rightarrow \phi_a(x)\rightarrow\phi_a(x)+\epsilon^{\nu}\partial_{\nu}\phi_a(x)$$ (where the minus sign is just due to it being an active transformation vs passive).

Similarly the Lagrangian transforms as $$\mathcal{L}\rightarrow\mathcal{L}+\epsilon^{\nu}\partial_{\nu}\mathcal{L}$$. Since the change is a total derivative we invoke Noether to get 4 conserved currents, one for each translation $$\epsilon^{\nu}$$:

$$(j^{\mu})_{\nu}=\frac{\partial\mathal{L}}{\partial(\partial_{\mu}\phi_a)}\partial_{\nu}\phi_a(x)-\delta^{\mu}_{\nu}\mathcal{L}\equiv T^{\mu}_{.\nu}$$

where T is the energy-momentum tensor satisying $$\partial_{\mu}T^{\mu}_{\nu}=0$$, i.e. conservation of energy-momentum for the classical field. But most importantly for this particular thread:

$$E=\int d^3xT^{00}$$
$$P^i=\int d^3xT^{0i}$$

i.e. $$P^{i}$$ is the Noether charge that is conserved arising from spatial translational invariance.

Now consider the most simple Lagrangian in QFT, the one leading to the KG equation:
$$\mathcal{L}=\tfrac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-\tfrac{1}{2}m^2\phi^2$$. You can easily plug this into the Noether current equation above to find:
$$T^{\mu\nu}=\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}$$

Therefore to find the classical momentum of this type of field:

$$P^i=\int d^3xT^{0i}$$
$$P^i=\int d^3x(\partial^{0}\phi\partial^{i}\phi-\eta^{0i}\mathcal{L})$$
$$P^i=\int d^3x(\partial^{0}\phi\partial^{i}\phi)$$

OK that was all classical, now if you quantize things, we turn P into an operator:

$$\vec{P}=-\int d^3x\pi\vec{\nabla}\phi=\int \frac{d^3p}{(2\pi)^3}\vec{p}a^{\dag}_{\vec{p}}a_{\vec{p}}$$

We can now take the state $$\mid p \rangle= a^{\dag}_{\vec{p}}\mid 0 \rangle$$, act on it with $$\vec{P}$$ to find

$$\vec{P}\mid p \rangle=p\mid p \rangle$$

and that's it, you can do exactly the same to find other operators using the other symmetries.

Like I said I'm newbie to all this, and I've basically just copied Tong's notes in the above, so I would appreciate any comment from Eugene or Frederik, on this too. Does it have limited scope? where do these mysterious "wavefunctionals" tie in with this picture? and how does what Eugene said about the operators being the old QM $$\hat{P}=-i\tfrac{\partial}{\partial x}$$ reconcile with this representation of the operators in field theory presented by Tong?

Cheers

 

[meopemuk]

$$\vec{P}=\int \frac{d^3p}{(2\pi)^3}\vec{p}a^{\dag}_{\vec{p}}a_{\vec{p}}$$

Yes, this is the correct form of the TOTAL momentum operator in the Fock space. If you apply this operator to any N-particle state you'll see that

$$\vec{P}= \vec{p}_1 + \vec{p}_2 + \vec{p}_3 + \ldots + p_N$$

So, your formula basically says that the total momentum of any system of particles is just the sum of individual particle momenta. In my opinion, there is no need to do field theory manipulations in order to arrive to this formula.

We can now take the state $$\mid p \rangle= a^{\dag}_{\vec{p}}\mid 0 \rangle$$, act on it with $$\vec{P}$$ to find

$$\vec{P}\mid p \rangle=p\mid p \rangle$$

and that's it, you can do exactly the same to find other operators using the other symmetries.

States $$a^{\dag}_{\vec{p}}\mid 0 \rangle$$ belong to the 1-particle sector of the Fock space. In this sector, the above total momentum operator $\vec{P}$ coincides with the 1-particle momentum operator. However, this coincidence is lost in higher N-particle sectors. In these sectors momentum operators of individual particles do not have connection with $\vec{P}$ (apart from the fact that the sum of momenta of all particles is equal to $\vec{P}$).

The observables of individual particles $$\vec{p}_i$$ cannot be easily expressed in terms of creation/annihilation operators. These observables act only within each separate N-particle sector. Each N-particle sector is exactly the same as the Hilbert space of usual N-particle quantum mechanics. All usual QM operator relationships remain valid there. This includes formula $$\hat{p}_i=-i\frac{\partial}{\partial x_i}$$ as well.


Eugene.

 

[Fredrik]

LAHLH: If you drop the interaction terms from the Lagrangian before you start this procedure, you will end up constructing a bunch of operators (generators of Lorentz transformations) that act on the Hilbert space of one-particle states. Then you can use that Hilbert space to construct the Hilbert space of many-particle states (the Fock space associated with the Hilbert space of one-particle states), and you can use the operators you constructed to construct the generators of Lorentz transformations on the Fock space.

I don't think this procedure works when there are interaction terms present (terms that contain products of more than 2 field components). My understanding (which really isn't that great...not yet anyway) is that all that stuff about the S-matrix and "in" and "out" states is a workaround that we're more or less forced to use because the procedure you're describing doesn't work when there are interaction terms.

I'm not familiar with the "wave functional" stuff that Samalkhaiat is talking about, so I can't really say anything about it.

 

[hamster143]

I don't recall seeing explicit forms of one-particle wave functionals or momentum operators in position space, but Weinberg presents the explicit form of Klein-Gordon vacuum wave functional in his book; thought I'd quote it here, just to give you an idea:

$$\Psi_{VAC}(\phi) = N \exp{(-\frac{1}{2} \int d^3 x d^3 y E(x,y) \phi(x) \phi(y))}$$

where N is some irrelevant normalization constant and

$$E(\vec{x},\vec{y}) = (2\pi)^{-3} \int d^3 p \exp{(i\vec{p}*(\vec{x}-\vec{y}))} \sqrt{p^2+m^2}$$

 

[samalkhaiat]

1) the field $\phi$ in $\Psi[\phi]$ is an ordinary function with the following transformation law

$$\bar{\phi}(\bar{x}) = D(a,A)\phi(x), \ \ A\in SL(2,\mathbb{C})$$

where D is a finite-dimentional, irreducible, non-unitary matrix representation of the Poicare' group.
The wave functional transforms as

$$\bar{\Psi}[\phi] = U(a,A)\Psi[D^{-1}\phi]$$

The representation U(a,A) is faithful, UNITARY and INFINITE-DIMENTIONAL but NOT IRRIDUCIBLE. So, we do have Poincare'-invariant norms (probabilities) in the Hilbert space of wave functionals.

2) QFT is based on quantization of a field theory on an arbitrary space-like surface. In this context we study transformations under an infinitesimal deformation of the quantization surface. The costomary way to apply this is by singling out a family of 3-dimensional surfaces and by introducing one evolution operator which transforms the state vector from one surface to the next one. the best known example is when we consider the family of plans t = constant. In this case, the Hamiltonian

$$H = \int d^{3}x \mathcal{H}= \int d^{3}x \ T^{00}$$

describes the evolution between different times. Please note that H does not drop down from the ski; it is simply the Noether charge associated with time-translation invariance.

The above quantization scheme is a general, PICTURE-INDEPENDENT scheme,i.e., it applies to both Schrodinger and Heisenberg representations of the states and operators.

3) In the COORDINATE representation of Schrodinger picture, we work with a basis for the Fock space where the (time-independent) field operator is diagonal:

$$\langle \bar{\phi}|\hat{\phi}(\vec{x})|\phi \rangle = \phi(\vec{x}) \delta (\phi - \bar{\phi}) \ \ (3.1)$$

This represents a field theory generalization of the familiar coordinate rep. in QM of point particle;

$$\langle \bar{x}|\hat{x}|x \rangle = x \delta(x-\bar{x})$$

Similarly, for the conjugate momentum, we have

$$\langle \bar{\phi}|\hat{\pi}(\vec{x})|\phi \rangle = -i \frac{\delta}{\delta \phi(\vec{x})} \delta (\phi - \bar{\phi}) \ \ (3.2)$$

which is a field theory analogue of

$$\langle \bar{x}|\hat{p}|x \rangle = -i \partial_{x} \delta (x - \bar{x})$$

Therefore,

$$t= 0 \ \ \ \ \ \ \ \ (3.3)$$

$$\hat{\phi}(\vec{x}) \rightarrow \phi(\vec{x}) \ \ \ (3.3a)$$

$$\hat{\pi}(\vec{x}) \rightarrow -i \frac{\delta}{\delta\phi(\vec{x})} \ \ (3.3b)$$

constitute a Schrodinger coordinate rep. of the equal time commutator algebra

$$[\hat{\phi}(x),\hat{\phi}(y)] = [\hat{\pi}(x), \hat{\pi}(y)] = 0$$

$$[\hat{\phi}(x),\hat{\pi}(y)] = i\delta^{3}(x - y)$$

Remember that in QM of point particle,

$$\hat{x}\rightarrow x$$
$$\hat{p}\rightarrow i\partial_{x}$$

is the coordinate rep. of the equal-time commutators,

$$[\hat{x}_{i},\hat{x}_{j}] = [\hat{p}_{i},\hat{p}_{j}] = 0$$
$$[\hat{x}_{i},\hat{p}_{j}] = i\delta_{ij}$$


4) In QM, we expand a state $|\psi(t)\rangle$ in the position basis $|x\rangle$;

$$|\psi(t)\rangle = \int \ dx \ \langle x|\psi(t) \rangle \ |x \rangle \ \ (4.1)$$

The component of $|\psi \rangle$ in the $|x \rangle$ -direction (which is just a number) is called wave function;

$$\psi(x,t) = \langle x|\psi(t) \rangle ,$$

is the amplitude for the system in state $|\psi \rangle$ to also be in the state $|x \rangle$.
When working with continuous systems (fields), eq(4.1) generalizes to

$$|\Psi(t) \rangle = \int \ D (\phi) \ \langle \phi | \Psi(t) \rangle \ \ |\phi \rangle \ \ (4.2)$$

The wave functional $\Psi[\phi,t] = \langle \phi |\Psi(t) \rangle$, represents the probability amplitude for the field to be in the configuration $\phi(\vec{x})$ at time t. Thus $|\Psi[\phi ,t]|^{2}$ is the probability density that the system will be found at the POINT $\phi$ in the field space F.
If F is the space of all maps:$\phi : \mathbb{R}^{3} \rightarrow \mathbb{R}$ or $\mathbb{C}$, from 3-dimensional space into the real or complex numbers, then a point $\phi \in F$ is a configuration of the field at a given instant. The wave functional $\Psi[\phi ,t]$ turns points in the field space F into real or complex numbers.

 

[samalkhaiat]

5) The state vector evolves according to the schrodinger equation

$$i\partial_{t} |\Psi(t) \rangle = H(\hat{\phi},\hat{\pi}) |\Psi(t) \rangle$$

Thus

$$i\partial_{t}\langle \phi |\Psi(t) \rangle = \int \ D(\bar{\phi}) \langle \phi |H|\bar{\phi} \rangle \langle \bar{\phi} |\Psi(t) \rangle$$

In the coordinate rep. the Hamiltonian is diagonal;

$$\langle \phi |H(\hat{\phi}.\hat{\pi})|\bar{\phi}\rangle = H(\phi , \frac{\delta}{\delta \phi}) \delta(\phi - \bar{\phi}) \ \ \ (5.1)$$

Thus the Schrodinger wave functional equation becomes

$$i\partial_{t}\Psi[\phi ,t] = H(\phi,\delta / \delta \phi) \Psi[\phi ,t] \ \ \ (5.2)$$

The formal solution of eq(5.2) is given by

$$\Psi[\phi,t] = \int \ D(\phi_{0}) \langle \phi |e^{-iHt}|\phi_{0}\rangle \Psi[\phi_{0},0] \ \ (5.3)$$

The time-evolution matrix element

$$G[\phi,t;\phi_{0},0] = \langle \phi |e^{-iHt}|\phi_{0}\rangle$$

represents the amplitude for the field to evolve from the configuration $\phi_{0}(\vec{x})$ at time t = 0 to the configuration $\phi(\vec{x})$ at t. So, the evolution is a "path" through a space made of copies of the field space F, stacked upon each other, each layer being labeled by the time $t \in \mathbb{R}$. That is, a point in the space $F \times \mathbb{R}$ is a mapping of spacetime into the real or complex numbers; $(\phi(\vec{x}),t) = \phi (\vec{x},t)$.

As in ordinary QM of point particle, we may write ( since H does not depend explicitly on the time);

$$\Psi[\phi,t] = e^{-iEt}\Psi[\phi] \ \ \ (5.4)$$

and find the time-independent Schrodinger equation

$$H(\phi,\frac{\delta}{\delta \phi})\Psi[\phi] = E \Psi[\phi] \ \ \ (5.5)$$

In the next post, I will show you how to solve eq(5.5) for the vacuum wave functional $\Psi_{0}[\phi]$ of the K-G field.

Until then, please see post #36 in
www.physicsforums.com/showthread.php?t=149261&page=3

also see posts 12, 14 and 16 in
www.physicsforums.com/showthread.php?t=180190

 

[samalkhaiat]

6) According to Noether theorem, the integral over the 3-space of the null component of the energy-momentum tensor defines the generator of infinitesimal translation; the 4-momentum operator:

$$P_{\mu} = \int d^{3}x \ T_{0\mu} = \int d^{3}x \left( \hat{\pi}(x)\partial_{\mu}\hat{\phi} - \eta_{0\mu}\mathcal{L} \right) \ \ \ (6.1)$$

where

$$\hat{\pi}(x) = \frac{\partial \mathcal{L}}{\partial \partial_{0}\hat{\phi}},$$

is the conjugate field momentum.

For free scalar field theory, eq(6.1) leads to the Hamiltonian

$$P_{0} = H(\hat{\phi}(x),\hat{\pi}(x)) = \frac{1}{2}\int d^{3}x \left( \hat{\pi}^{2} + |\nabla\hat{\phi}|^{2} + m^{2}\hat{\phi}^{2}\right), \ \ \ (6.2)$$

and 3-momentum operator

$$P_{j} = - \int d^{3}x \hat{\phi}(x) \partial_{j} \hat{\pi}(x) \ \ \ (6.3)$$

In the coordinate Schrodinger representation, H and P become time-independent functional differential operators:

$$H = \frac{1}{2} \int d^{3}x \left( -\frac{\delta^{2}}{\delta \phi^{2}(\vec{x})} +|\nabla \phi(\vec{x})|^{2} + m^{2} \phi^{2}(\vec{x})\right) \ \ \ (6.4)$$

$$P_{j} = i \int d^{3}x \phi(\vec{x}) \partial_{j} \frac{\delta}{\delta \phi(\vec{x})} \ \ \ (6.5)$$

Thus the time-independent Schrodinger equation, eq(5.5), becomes

$$\frac{1}{2} \int d^{3}x \left( - \frac{\delta^{2}}{\delta \phi^{2}(\vec{x})} + |\nabla \phi|^{2} + m^{2}\phi^{2}\right) \Psi[\phi] = E \Psi[\phi] \ \ (6.6)$$

Since we are interested in the lowest energy solution, $\left( E_{0}, \Psi_{0}[\phi]\right)$, we may assume that $\Psi_{0}[\phi]$ has no nodes and positive everywhere. Thus, we can write

$$\Psi_{0}[\phi] = C \exp (- G[\phi]) \ \ \ (6.7)$$

where C is a constant determined by the usual mormalization,

$$\int \ D(\phi) |\Psi[\phi]|^{2} = 1,$$

and the functional $G[\phi]$ satisfies

$$\frac{1}{2}\int d^{3}x \left( \frac{\delta^{2}G}{\delta \phi^{2}} - (\frac{\delta G}{\delta\phi})^{2} \right) = E_{0} - \frac{1}{2} \int d^{3}x \phi(\vec{x}) (- \nabla^{2} + m^{2})\phi(\vec{x}) \ \ (6.8)$$

Let us write,

$$G[\phi] = \int d^{3}x d^{3}y \phi(\vec{x})g(\vec{x},\vec{y})\phi(\vec{y}), \ \ \ (6.9)$$

and (to make life easy) take the unknown function g(x,y) to be symmertic in x and y. Inserting eq(6.9) back into eq(6.8) and equating equal powers terms, we find

$$E_{0} = \int d^{3}x g(\vec{x},\vec{x}) \ \ \ (6.10)$$

and

$$(- \nabla^{2} + m^{2}) \delta^{3}(z-y) = \int d^{3}x g(\vec{z},\vec{x})g(\vec{x},\vec{y}) \ \ \ (6.11)$$

The left-hand side of eq(6.11) suggests that the right-hand side must only depend on $(z - y)$.
Thus, we may assume that g is translationally invariant and represent it by the following Fourier integral

$$g(\vec{x},\vec{y}) = g(\vec{x}-\vec{y}) = \int \frac{d^{3}k}{(2\pi)^{3}} \bar{g}(\vec{k}) e^{i\vec{k}.(\vex{x}-\vec{y})}$$

The Fourier transform of eq(6.11) leads to

$$\bar{g}^{2}(\vec{k}) = (1/4)(k^{2} + m^{2}) = \frac{1}{4}\omega_{k}^{2}$$

Therefore, the unknown function g is given by

$$g(x,y) = \frac{1}{2} \int \frac{d^{3}k}{(2\pi)^{3}} \omega_{k} e^{i\vec{k}.(\vec{x}-\vec{y})} \ \ (6.12)$$

Putting eq(6.12) in eq(6,10) yields the vacuum state energy

$$E_{0} = \frac{1}{2} \int d^{3}k \ \omega_{k} \ \delta^{3}(0)$$

I am sure you do recognize this divergent result, don't you? It is exactly the result we obtain in the usual (Heisenberg) operator formalism of QFT.
You can also show that the vacuum carries no momentum:

$$P_{j}\Psi[\phi] = 0$$

Exercises for you:

Show that the 1st excited state wave functional is given by

$$\Psi_{1}[\phi] = (\frac{2\omega_{k_{1}}}{(2\pi)^{3}})^{1/2} \int d^{3}x e^{-ik_{1}.x} \phi(\vec{x})\Psi_{0}[\Phi]$$

Also show that $\Psi_{1}$ is a momentum eigenstate

$$P_{j}\Psi_{1}[\phi] = (\vec{k_{1}})_{j} \Psi_{1}[\phi]$$

Since $\Psi_{1}$ is also an energy eigenstate with energy $\omega_{k_{1}}$ relative to the vacuum, we can use $\Psi_{1}[\phi]$ as a state describing one particle with 4-momentum $k_{1}$ and mass m.

Another exercise for you

Transform the creation and destruction operators of the real scalar field into the coordinate Schrodinger representation, and show that:

$$a(\vec{k}) \Psi_{0}[\phi] = 0$$

$$a^{\dagger}(\vec{k}_{1})\Psi_{0}[\phi] = (2\omega_{k_{1}}(2\pi)^{3})^{1/2}\Psi_{1}[\phi]$$


regards to you all

sam

 

[JustinLevy]

I really want to understand this, but I think I'm missing something fundamental here. I don't understand how what you wrote corresponds to the creation and annihilation operators.

In particular, you write:

$$\hat{\phi}(\vec{x}) \rightarrow \phi(\vec{x}) \ \ \ (3.3a)$$

What happenned to Fock space? Shouldn't this be a function of all the particle positions? Why is it only a function of a single spacetime coordinate?

I would naively expect the functional to be a functional taking a set of "field functions" for each possible number of particles -> a complex number. No?

I really want to understand this better. It would also help me answer my own question here:
https://www.physicsforums.com/showthread.php?t=390519

 

[hamster143]

I really want to understand this, but I think I'm missing zsomething fundamental here. I don't understand how what you wrote corresponds to the creation and annihilation operators.

In particular, you write:

What happenned to Fock space? Shouldn't this be a function of all the particle positions? Why is it only a function of a single spacetime coordinate?

I would naively expect the functional to be a functional taking a set of "field functions" for each possible number of particles -> a complex number. No?

I really want to understand this better. It would also help me answer my own question here:
https://www.physicsforums.com/showthread.php?t=390519

In single-particle QM, we have$$\hat{x}\rightarrow x$$
$$\hat{p}\rightarrow -i\partial_{x}$$

which means that

$$\hat{x} \Psi(x) = x \Psi(x)$$
$$\hat{p} \Psi(x) = -i\partial_{x} \Psi(x)$$

Equations 3.3a/b say that, in QFT, these become

$$\hat{\phi} \Psi[\phi(\vec{x})] = \phi(\vec{x}) \Psi[\phi(\vec{x})]$$
$$\hat{\pi} \Psi[\phi(\vec{x})] = -i \frac{\delta}{\delta\phi(\vec{x})} \Psi[\phi(\vec{x})]$$

 

[samalkhaiat]

I really want to understand this, but I think I'm missing something fundamental here. I don't understand how what you wrote corresponds to the creation and annihilation operators.

Simply write the creation and annihilation operators of the K-G field in terms of $\phi(x)$ and $\pi(x)$, then put

$$x^{0}=t=0$$

and then substitute the operator $\hat{\phi}(\vec{x},0)$ by its eigen value $\phi(\vec{x})$, i.e.,

$$\hat{\phi}(\vec{x})|\phi> = \phi(\vec{x})|\phi>$$

and the operator $\hat{\pi}(\vec{x})$, by its functional differential realization,i.e.,

$$\hat{\pi}(\vec{x}) = -i \frac{\delta}{\delta \phi(\vec{x})}$$

You will then get

$$a(\vec{k}) = \int d^{3}x e^{i\vec{k}.\vec{x}}\left( \omega_{k}\phi(\vec{x}) + \delta / \delta \phi(\vec{x}) \right)$$

Shouldn't this be a function of all the particle positions?

we do not speak of particle or particles when we deal with the field FUNCTION.

Why is it only a function of a single spacetime coordinate?

THERE IS NO SPACETIME COORDINATE, we deal with 3-space coordinates x,y,z.

I would naively expect the functional to be a functional taking a set of "field functions" for each possible number of particles -> a complex number. No?

It is true when and only when you delet "for each possible number of particle" from your statement.

regards

sam

 

[meopemuk]

The wave functional $\Psi[\phi,t] = \langle \phi |\Psi(t) \rangle$, represents the probability amplitude for the field to be in the configuration $\phi(\vec{x})$ at time t. Thus $|\Psi[\phi ,t]|^{2}$ is the probability density that the system will be found at the POINT $\phi$ in the field space F.

What is the operational (experimental) definition of this "probability"? Experimentalists know pretty well how to measure particle observables (position, momentum, spin, etc.) and how to define associated probabilities. What is your advise for experimentalists about how to measure "the probability density for the field to be in the configuration $\phi(\vec{x})$ at time t"?

Eugene.

 

[JustinLevy]

I still don't understand why it is only a function of one position. This seems to suggest to me you don't have freedom to describe correlations between two particle positions, since the particle density alone would provide a restriction for every point ... and trying to specify the particle correlations as well would then over restrict the solution?

I'm clearly still missing something here.

Let's consider a more concrete example.
For example, say I wanted to look at the bound states of an electron and a positron. Is the functional now map three fields for an electron and a positron and photons (and each field being just a function of a spatial coordinate) to a single complex number?

What would the eigen value equation look like for this functional? ... I don't understand how you go from the lagrangian with the fields as the degrees of freedom, to getting evolution of a functional of the fields. Where does the functional come from? Its not even in the lagrangian. You seem to be saying that it is the functional that is the state information and is evolving though ... the functional is the degree of freedom, and the fields are just the coordinates to describe the functional. I don't see how these all fit together.

I'm sorry I'm asking so many questions. I really want to understand this, and something just isn't clicking.

 

[Physics Monkey]

Perhaps it would help to regulate the problem. Think of the field theory with a cutoff so that one has a finite number of degrees of freedom per unit volume. For a scalar field we might think of having a single real degree of freedom $$q_x$$ for each point $$x$$ in some lattice. Processes on length scales long compared to the lattice spacing are well described by the continuum field theory.

This collection of a finite number of degrees of freedom has a wavefunction $$\Psi(\{ q_x \} )$$ which is a function of the values of all the local degrees of freedom. For a collection of oscillators (like a free theory) this wavefunction will be quadratic in the degrees of freedom $$\Psi \sim \exp{\left( - \frac{1}{2} \sum_{x y} q_x K_{x y} \,q_y \right) }$$. The wavefunction maps a particular configuration of all the $$q_x$$ into a single complex number.

The field theory wave function is conceptually identical. The only difference is that the discrete set $$\{ q_x \}$$ is replaced by a continuous set $$\{ q(x) \}$$ of variables. The wavefunction takes in a particular field configuration at fixed time and outputs a complex number, the amplitude for that configuration. One doesn't have to speak about particles, but the information is in there. For example, the wavefunction i.e. the quantum state in the "field eigenbasis" is related by a change of basis to the state in the Fock basis.

Does this help at all?

 

[JustinLevy]

Does this help at all?

That makes intuitive sense, until this:

...One doesn't have to speak about particles, but the information is in there. For example, the wavefunction i.e. the quantum state in the "field eigenbasis" is related by a change of basis to the state in the Fock basis.

I get lost there.
In the Fock basis, there would be a function of N positions for the N particle basis, right? And then there'd be a sum over all N.

This seems to allow correlations between particle positions that don't seem possible to me with a function of only one position.

Could someone please show the steps connecting the state in the "field eigenbasis" to the state in the "Fock basis" a bit more explicitly? Not just the math of "here is an equation", but the procedure ... ie. how we derive the connection between the two. Because as soon as we try to add interactions, I have a feeling this is going to get even more confusing. And how are antiparticles represented?

Can someone show what the "wavefunctional" equation would be for QED? I still don't understand how you are deriving these things since there are fields in the Lagrangian, but no wavefunctional.