Strings vs. point interactions in QFT?

[Gedankenspiel]

Hi all!

I have a question regarding the principal difference between QFT and string theory according to popular accounts. It is said that QFT deals with point particles leading to the well-known infinities in calculating the transition amplitudes whereas in string theory the interaction is smeared out and no infinities occur.

I don't understand this argument: QFT is the quantum theory of fields whereas string theory is the quantum theory of strings. The world sheet has only two dimensions and seems to be more singular than a field, which extends in four dimensions. So there would be less reason for infinities to occur in QFT than in string theory according to this argument.

How can we talk about point particles in QFT anyway when it is a quantum theory of fields?

 

[haushofer]

Just a quickie: QFT is a field theory where particles are excitations of fields, whereas string theory describes excitations of strings. This is not string field theory; the strings themselves are not created and annihilated.

So you should compare the excitations of QFT, the 'particles', with strings, so 0-dim. objects versus 1-dim objects.

 

[Demystifier]

A good way to explain this is comparison with the books Bjorken-Drell 1 and Bjorken-Drell 2. In Bjorken-Drell 1 everything, including Feynman diagrams, is expressed in a language of first quantization, where starting objects are particles rather than fields. Only in Bjorken-Drell 2 all this is reformulated in terms of fields. In string theory, however, all books can be thought of as generalizations of Bjorken-Drell 1.* We still don't have Bjorken-Drell 2 for strings.**

*Books also talk about M-theory, but this is not really a field theory.
**Books do say something about string-field theory, but this topic is not very well understood except for unrealistic bosonic strings.

 

[Gedankenspiel]

Thinking about it, I am wondering more about why we are entitled to talk about point particles in QFT (which is usually done, I guess). The theory is about field excitations. Where's the connection to point-like particles?

 

[haushofer]

Well, particles are characterized by their quantum numbers and mass, but not by some "volume".

 

[Urs Schreiber]

The key to understanding how perturbative string theory relates to QFT is to adopt the "worldline formalism"-perspective on QFT, see here, and see string theory FAQ – What is the relationship between quantum field theory and string theory?

Traditionally the perturbation series/scattering amplitudes of a quantum field theory are defined, given an action functional/local Lagrangian, by applying the Feynman rules to the monomial terms in the Lagrangian and deriving from that (via “second quantization”) a rule for how to weight each Feynman diagram by a probability amplitude.

In what is called the worldline formalism of quantum field theory this assignment is obtained instead more conceptually as the correlators/n-point functions of a 1-dimensional QFT that lives on the graphs, namely the worldline theory (usually a sigma-model in the given target spacetime) of the particles that are the quanta of the fields in the field theory.

Mathematically the key step here is a Mellin transform – introducing a “Schwinger parameter” – which turns the partition function of a worldline theory, schematically of the form $t \mapsto Tr\, \exp(-t H)$ (for $H$ the Hamiltonian/wave operator) into the zeta regulated Feynman propagator

$$
\hat \zeta_H(s) = \int_0^\infty Tr_{reg} \exp(-t H) t^{s-1} d t = Tr_{reg} H^{-s}
\,.
$$

This worldline formalism is equivalent to the traditional formulation. It has the conceptual advantage that it expresses the QFT (in perturbation theory) more manifestly as a second quantization of its particle content given explicitly of the superposition of all 1-particle processes.

The worldline formulation of QFT has an evident generalization to higher dimensional worldvolumes: in direct analogy one can consider summing the correlators/n-point functions over worldvolume theories of “higher dimensional particles” (“branes”) over all possible worldvolume geometries. Indeed, for 2-dimensional branes this is precisely the way in which perturbative string theory is defined: the string scattering amplitudes are given by the analogous “worldsheet formalism” known as the string perturbation series as the sum over all surfaces of the correlators/n-point functions of of a 2d SCFT of central charge 15.
https://dl.dropboxusercontent.com/u/12630719/WorlvolumeFormalism.JPG

Indeed, after decades of Feynman rules, the worldline formalism for QFT was found only via string theory in (Bern-Kosower 91), by looking at the point particle limit of string scattering amplitudes. Then (Strassler 92, Strassler 93) observed that generally the worldline formlism is obtained from the correlators of the 1d QFT of relativistic particles on their worldline.

(from (Schmidt-Schubert 94))

 

[Demystifier]

The theory is about field excitations. Where's the connection to point-like particles?

The excitations have a discrete spectrum characterized by an integer $n$. For given $n$, the corresponding quantum state can be represented by a wave function $\psi(x_1,\ldots,x_n)$. This has all mathematical properties of the $n$-particle wave function familiar from quantum mechanics of particles. In particular, in the non-relativistic limit, $|\psi(x_1,\ldots,x_n)|^2$ can be interpreted as probability density of finding particles at the positions $x_1,\ldots,x_n$.

 

[Gedankenspiel]

Sorry for the long delay, unfortunately I was lacking the time to answer.

So, judging from your answers, I am wondering if QFT really is what it claims to be: a quantum theory of fields. This would mean that a state in QFT is a functional of classical field configurations. Its squared modulus would be a probability distribution giving the probability density for measuring a certain classical field configuration.
Instead, for the state space of QFT one only reads about Fock space, which is essentially derived from the one-particle state space of a point particle and whose elements are superpositions of multi-particle states.

Are these two spaces in fact the same or not? Where is the connection made explicit, between the Fock space and the space of complex functionals over over classical field configurations? Is there any, for that matter?

 

[Demystifier]

So, judging from your answers, I am wondering if QFT really is what it claims to be: a quantum theory of fields.

That's a good question, asked also in
https://arxiv.org/abs/0705.3542

Are these two spaces in fact the same or not? Where is the connection made explicit, between the Fock space and the space of complex functionals over over classical field configurations? Is there any, for that matter?

Yes, they are the same spaces. For the connection see e.g.
https://arxiv.org/abs/quant-ph/0208185
Sec. 3.1 and references therein.