Path Integral in first and second quantization

[friend]

Is it true that in first quantization the PI includes the possible trajectories a particle can take, but it does not include how particles can change into other kinds of particles (electrons to photons, etc). And QFT (second quantization) calculates how particles can branch off into other particles, but it does not calculate their trajectories. Is this right? Thanks.

 

[jfy4]

The sum over histories part are all the probabilities of the particle traversing the different paths, not the particle actually traveling them. The probabilities of the paths that are far from the extrema oscillate wildly and hence over integration go to zero under the stationary phase approximation.

 

[tom.stoer]

Roughlyspeaking the PI in QM is a sum over all trajectories in x-space, whereas the PI in QFT is sum over all field configurations.

 

[friend]

Roughlyspeaking the PI in QM is a sum over all trajectories in x-space, whereas the PI in QFT is sum over all field configurations.

By "field configurations" you mean particle states, right?

I suppose there is no integral that include both trajectories and field configurations. The PI of trajectories is integrated over line segments, whereas the QFT PI integrate over area segments, right? So we use each PI in its own context and they cannot be mixed, right?

 

[tom.stoer]

No, I mean field configurations, not "states".

PI in QM

Suppose you have a particle with a trajectory x(t). You take all trajectories x(t) and calculate their action S[x(t)]. Then you "sum" over all these contributions, i.e. you calculate the "interference between these trajectories using the weight exp[iS].

If the particle lives in 3 dimensions you have to introduce an index i=1..3 like x_i(t) which means that the action is something like S[x_i(t)]

PI in QFT

For fields the situation is different. Suppose you have a field A(x,t) for each spacetime point (x,t). First not that there is no x(t) anymore. To see how the PI in QFT is related to the PI in QM we re-write A(x,t) as A_x(t) where x is now a "continuous index.

Now we have to calculate the action for all possible field configurations, i.e. S[A_x(t)]. Instead of "summing" over all posssible x_i(t) we sum over all possible A_x(t) with the continuous index x.

I hope this makes clear why there is no "path" x(t) anymore. x is nothing else but an index. Please note the increase in complexity when going from x_i(t) to A_x(t)

 

[friend]

I suppose there is no integral that include both trajectories and field configurations. The PI of trajectories is integrated over line segments, whereas the QFT PI integrate over area segments, right? So we use each PI in its own context and they cannot be mixed, right?

Asked differently, I wonder how does a particle "propagate" through a quantum field? Is there any relation between what configuration a field has at a particular point in spacetime to the configuration that field has a small distance away?

 

[tom.stoer]

I see no chance to derive something like a "particle" from the QFT PI.

 

[vanhees71]

The question is, what you mean by "derive particles from QFT" (no matter whether in the Hilbert-space or the path-integral formulation). From the very beginning, you consider QFT as a description of particles (if applied to elementary-particle physics ;-)).

So the question is, what do you define as a particle. Let's look at another application of QFT, namely many-body theory (be it non-relativistic as in most condensed-matter applications or relativistic as in heavy-ion physics). Then you describe the single-particle properties of the many-body system (you see, also here we start from the idea of particles from the very beginning!) with help of the two-point in-medium Green's function (i.e., the propagator in the Schwinger-Keldysh real-time formulation). As it turns out, in many cases, that the spectral function associated with this Green's function has very narrow peaks as function of energy at fixed momentum. Then, you have something which is very close to free particles in the vacuum, because the Green's function of a free particle in the vacuum just has a pole at the on-shell point $p_0=\sqrt{\vec{p}^2+m^2}$ (for relativistic particles). Thus, one has a situation, where you can think in terms of "particles" when considering the one-particle excitations of the medium. Usually, these "particles" have different properties than the particles in the vacuum, with which you started with. E.g., in condensed matter physics very often you can build an effective theory of fermions in the medium which have the same quantum numbers as electrons but have a larger mass ("heavy electrons"). In other cases you may find excitations with totally different properties than the particles you started with. E.g., in solid-state physics of metals you start with a lattice of positively charged ions and some more or less freely moving electrons and you find not only electron-like degrees of freedom (perhaps with modified masses) but also collective modes corresponding to the quantized vibrations of the crystal lattice. These quasiparticles are called phonons, and an effective model of interacting electrons and phonons often can be a good description of the metal.

Sometimes you find very exotic things, when investigating many-body systems. E.g., in certain materials, called "spin ice", you find collective modes that can be described by quasiparticles that behave like Dirac's magnetic monopoles. Or take the phantastic material, graphen, where you have a single layer of graphite, for which the electrons are effectively bound to a plane. Calculating the quasiparticle excitations of these electrons leads to quasiparticles that behave like massless Dirac fermions in two dimensions (among other very exciting things).