Undergrad The construction of particles in QFT

[vanhees71]

Math in QFT is bringing much artificial renormalizations in the end and enormous calculations. Surely the nature does it not that way.
I have a question apart from mathematical formulation. Is the QFT field a field of probabilities? Operator valued field is a field in math but what I want to know is what is it in reality? The collapse is possible only for probabilities.

Well, renormalization is primarily not about the divergences, which only occur, because physicists tend to handle distribution (generalized functions) sloppily. You have to renormalize in any case when doing any perturbation theory, and you can indeed do the calculations of perturbative QFT without any divergences occurring at any stage. You just have to handle the distributions, particularly taking products of them, carefully.

The traditional approach is to do some regularization of the distributions first and then taking limits to the physical parameters after renormalization, which by construction is finite. For perturbative calculations the most convenient techniques of this kind is dimensional regularization or the zeta-function renormalization, which is closely related to Schwinger's proper-time method.

Another systematic scheme is to interpret the naive Feynman rules, leading to divergent "loop integrals", as rules for the integrands of the expression expressed by the diagrams and then doing the necessary subtractions, leading to the renormalized expressions for these integrands and integrating the (then of course finite) expressions over the loop momenta. This technique also works for resummed perturbation theory (like the 2PI/CJT formalism).

An alternative more modern approach is the "causal Epstein-Glaser" approach, where you work with physical fields, i.e., normalizable states and not with distributions. This is a very physical idea.

Last but not least there's also the functional renormalization-group approach, which corresponds to the most physical approach towards renormalization, which is Wilson's approach.

All these techniques to remedy the UV divergences lead to the same result for a given renormalization scheme, and all these schemes lead to the same physical results (S-matrix elements) to the order of perturbation theory calculated.

If massless fields are involved in the theory (as is the case for all gauge theories with un-Higgsed local gauge theories) you also have to take care about infrared and colinear divergences, which is done traditionally by summing up soft-photon contributions all contributing at the same order of the expansion parameter (powers of the coupling constant or rather powers of $\hbar$). A more modern (equivalent) approach is to work with the correct "asymptotic free states", which are not the naive plane waves usually used but a specific kind of "dressed states" taking into account the non-trivial asymptotics due to the long-range nature of gauge interactions (socalled "infra-particles").

 

[A. Neumaier]

But "plus/minus infinity" in the QFT model represent "detection at the detector/emission at the source" in the actual experiment. So they are how QFT represents collapse, since those events are where collapse occurs in ordinary QM.

But this collapse is outside of the QFT machinery - unlike in Dirac's collapse, nothing is said and can be said in QFT about the state after what you call the QFT analog of the collapse. Thus there is no collapse in QFT proper.

 

[mattt]

@mattt Where does your attachment in post #26 come from? If there is a link to it online, you should give the link, not post it as an attachment.

I'm sorry, I wasn't aware of this rule. I'll fix it.

EDIT: I think I took it from here:

http://www.math.lmu.de/index.html

but now I can't find it. Anyway, it was just one example ( to show what a Quantum field is mathematically ).

 

[Demystifier]

I think QM is about waves (ascribed to objects regarded poinlike). In Copenhagen interpretation particles don't have positions. So...

So you never heard of position operator in QM? If so, I would suggest you to first learn more about QM before trying to understand QFT.

 

[ilper]

An example of a real valued distribution is Dirac's delta function.

Apart from that I could not imagine what would that mean physically (an operator acting on every state and giving infinity) but math gives every imaginary possible.
Ok but what do we get when measured. Any number of particles with a subsequent probability, or what?

 

[ilper]

So you never heard of position operator in QM? If so, I would suggest you to first learn more about QM before trying to understand QFT.

Having position operator is not the same as having position. There is no position operator with eigenfunction Dirac delta function.

 

[Demystifier]

Having position operator is not the same as having position. There is no position operator with eigenfunction Dirac delta function.

True, but the same can be said for field operator in QFT.

 

[ilper]

No. QFT is not modeling the process as "things in space changing in time". It is modeling the process as a whole self-consistent 4-dimensional spacetime with self-consistent 4-dimensional objects in it. There is no "vanishing" of anything at the end.

That's for sure. But in order to grasp things physically as a mental picture it is not so bad. Zee in his book is adopting it. Note also that there is a nonrelativistic variant of QFT.
I am more interested in your note "Nothing is vanishing at the end." If so there is no collapse and one would expect the excitation to be measured in other points as well?

 

[A. Neumaier]

Apart from that I could not imagine what would that mean physically (an operator acting on every state and giving infinity) but math gives every imaginary possible.
Ok but what do we get when measured. Any number of particles with a subsequent probability, or what?

Measured is the expectation value of operators obtained by integrating the quantum field over a finite region of spacetime. This produces extensive numbers (with a computable uncertainty) roughly proportional to the integrated volume, not probabilities. Just like measuring the concentration of salt at various places in a lake produces extensive numbers, not probabilities.

 

[ilper]

Measured is the expectation value of operators obtained by integrating the quantum field over a finite region of spacetime. This produces extensive numbers (with a computable uncertainty) roughly proportional to the integrated volume, not probabilities. Just like measuring the concentration of salt at various places in a lake produces extensive numbers, not probabilities.

Ha, to get the expectation value you need the probabilities for every value from the eigenvalues of the operator. So it is what I thought - loosely speaking a couple of vectors from eigenvalues and probabilities in spacetime.

 

[A. Neumaier]

Ha, to get the expectation value you need the probabilities for every value from the eigenvalues of the operator. So it is what I thought - loosely speaking a couple of vectors from eigenvalues and probabilities in spacetime.

No. To predict the expectation value you just use the formula $\langle A\rangle=Tr~\rho A$. The measurement does not produce numbers with probabilities for averaging; it produces directly the expectation value.

 

[Demystifier]

No. To predict the expectation value you just use the formula $\langle A\rangle=Tr~\rho A$. The measurement does not produce numbers with probabilities for averaging; it produces directly the expectation value.

But it doesn't get rid of probabilities. To account for the possibility of different measurement outcomes, your $\rho$ needs to satisfy a stochastic equation, which, of course, involves probabilities.

 

[PeterDonis]

If so there is no collapse and one would expect the excitation to be measured in other points as well?

You're missing the point. When you define a model in QFT, you are defining in advance exactly where in spacetime measurements are being made; those are the "sinks". If you change where measurements are made, you change the model, and you have to re-do all your calculations with the changed model. Asking whether something could be measured at other places than where it is measured in a given QFT model makes no sense.

 

[ilper]

You're missing the point. When you define a model in QFT, you are defining in advance exactly where in spacetime measurements are being made; those are the "sinks". If you change where measurements are made, you change the model, and you have to re-do all your calculations with the changed model. Asking whether something could be measured at other places than where it is measured in a given QFT model makes no sense.

Does the light from a star know you will measure it on Earth? If you doesn't intend to measure it it will not shine?
If QFT is just a model so pure defined, where do come from this talks about fields being building blocks of Universe from David Tong (not sure about his name but I saw a lecture in Cambridge or Oxford on youtube).

 

[PeterDonis]

Does the light from a star know you will measure it on Earth?

You're missing the point again. When you use QFT to model a photon from a distant star being received on Earth, those two events (emission from the star, detection on Earth) are inputs to the model. You already know that those are the endpoints. You don't use QFT to find that out.

If you want to use QFT to find out where the light from the distant star might go, you use multiple models, choosing different endpoints for each one. For example, you could pick the Earth as the endpoint in one model and some planet in the Andromeda galaxy as the endpoint in another. Then those two calculations would give you the relative probability of a photon from the distant star being detected at those two locations.

where do come from this talks about fields being building blocks of Universe from David Tong

From the fact that the most fundamental theory we have of everything except gravity is a QFT--the Standard Model of particle physics. None of which changes anything I said.

 

[A. Neumaier]

But it doesn't get rid of probabilities. To account for the possibility of different measurement outcomes, your $\rho$ needs to satisfy a stochastic equation, which, of course, involves probabilities.

My statement was about measuring quantum fields. The only statistics appearing is the same as that for measuring classical fields as in pre-quantum times. Born's rule is not involved.

 

[ilper]

From the fact that the most fundamental theory we have of everything except gravity is a QFT--the Standard Model of particle physics. None of which changes anything I said.

In QM (intended to be TOE) things don't happen just for separate cases. And QFT is following QM making it relativistic.

 

[EPR]

In QM (intended to be TOE) things don't happen just for separate cases. And QFT is following QM making it relativistic.

Those 'separate' cases form the observed reality.

 

[ilper]

Those 'separate' cases form the observed reality.

Mathematics is for the separate cases. Physics is about the whole physics (nature).

 

[EPR]

Nature appears to be fundamentally probabilistic. This is the new physics.

 

[ilper]

Those 'separate' cases form the observed reality.

Nature appears to be fundamentally probabilistic. This is the new physics.

Well but you see what David Tong is saying that QF are building blocks. That does not fit with probability.

 

[ilper]

If you want to use QFT to find out where the light from the distant star might go, you use multiple models, choosing different endpoints for each one. For example, you could pick the Earth as the endpoint in one model and some planet in the Andromeda galaxy as the endpoint in another. Then those two calculations would give you the relative probability of a photon from the distant star being detected at those two locations.

You insisted earlier that in QFT nothing vanishes. But if I get the photon on Earth the poor creatures from Andromeda will not detect the excitation of the field nevertheless that in the multiple model it has reach them in order to calculate the relative probability. The only decision one is left is that the field is a probability and the instant I get the photon the field (vanishes there) returning to vacuum state. (like Born 'solution' to QM paradoxes).

 

[ilper]

My statement was about measuring quantum fields. The only statistics appearing is the same as that for measuring classical fields as in pre-quantum times. Born's rule is not involved.

If the the statistic is as for classical fields you will receive a definite result and expectation value is superfluous. Then so are also operators.

 

[A. Neumaier]

If the statistic is as for classical fields you will receive a definite result and expectation value is superfluous. Then so are also operators.

The ensemble expectation values of the quantum fields are the quantum versions of the values of the classical fields. They don't have a statistical interpretation - one cannot prepare multiple copies of a quantum field at the same spacetime location to get its statistics!

 

[vanhees71]

Well but you see what David Tong is saying that QF are building blocks. That does not fit with probability.

Quantum fields use the concepts of quantum theory, and quantum theory is intrinsically probabilistic, and as far as we know for more than 90 years now Nature seems to be intrinsically probabilistic. Many people seem still to have some quibbles with this, because of philosophical prejudices. This brought, however, some of the most profound findings of modern physics, particularly in quantum optics, and this lead to the newest generation of engineering science, i.e., quantum-information technology.

You may still have your philosophical prejudices against a probabilistic nature, but the facts are the facts!

 

[ilper]

Quantum fields use the concepts of quantum theory, and quantum theory is intrinsically probabilistic, and as far as we know for more than 90 years now Nature seems to be intrinsically probabilistic. Many people seem still to have some quibbles with this, because of philosophical prejudices. This brought, however, some of the most profound findings of modern physics, particularly in quantum optics, and this lead to the newest generation of engineering science, i.e., quantum-information technology.

You may still have your philosophical prejudices against a probabilistic nature, but the facts are the facts!

So Quantum Fields are probabilistic fields and not some real stuff.
1. one has different probabilities for the eigenvalues of the operator of some physical quantities in spacetime.
2. after measurement they collapse (to vacuum state) as its cousin the wavefunction (to 0)
Do you agree?

 

[EPR]

What do you mean by 'real'?
Quantum fields are matter - what you observe in daily life(basically fermionic and bosonic fields).

Is an electron real? Certanly not classically real but a very measureable entity.

 

[vanhees71]

No, of course not.

In quantum theory you have a Hilbert space and an algebra of observables realized as self-adjoint operators on Hilbert space the eigenvalues of these operators are the values the observables described by them can take. Then there is a self-adjoint operator with trace 1, the statistical operator, that describes the state of the system and is determined by the preparation of the described system at the initial time of the experiment. With the eigenvectors of the observable operators and the statistical operator you can calculate the probability to find any of the possible values of an observable when measured, given the state of the system. There not more to this. In general you cannot say what happens to the system in a measurement. This you have to analyse given the concrete setup of the measurement. The socalled collapse of the state is an ad-hoc assumption of some kinds of the Copenhagen interpretation and inconsistent with relativistic local quantum field theory, according to which there cannot be any instantaneous actions at a distance by assumption. Fortunately the collapse assumption is not needed to describe nature with quantum theory.

In relativistic quantum field theory the observable operators are constructed with help of fundamental field operators (which usually do not directly represent observables). This is a mathematical concept to construct locally interacting field theories which are consistent with the relativistic space-time description and its implied notion of causality.

The most common experiment described by relativistic QFT are scattering experiments, which consist in the preparation of two particles (described by asymptotic free "in-states") which scatter at each other when coming close. What's then observed after the collision are again particles (described by asymptotic free "out-states") and their properties (mass- and momentum distributions etc.), i.e., you describe a reaction in collisions in terms of cross sections, which are transition probability rates per incoming-particle flux for a given reaction.

 

[ilper]

What do you mean by 'real'?
Quantum fields are matter - what you observe in daily life(basically fermionic and bosonic fields).

Is an electron real? Certanly not classically real but a very measureable entity.

I can not tell what is real but I can tell what is not. The probability is not real. Then as the fields are probabilities there are also not real. And the electron which is an excitation of probability is not real. Nothing is real.

 

[A. Neumaier]

I can not tell what is real but I can tell what is not. The probability is not real. Then as the fields are probabilities there are also not real. And the electron which is an excitation of probability is not real. Nothing is real.

Fields are not probabilities. You are thoroughly mistaken in your views.