Who is Ballentine and why is he important in the world of quantum mechanics?

  • #1
joneall
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People ask me what quantum fields are. (Really...) Quantum fields are the basic stuff of physical (and all) reality, right? But, after second quantization, they are operators! I cannot get my head around this.
 
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  • #2
It seems that you are mixing the physical entity and its mathemarical counter part, because they are called by the same name.
 
  • #3
Sorry, isn't the math supposed to represent physical entities? Is there a distinction I haven't grasped. (In fact, there's a lot I haven't grasped.)
 
  • #4
joneall said:
Quantum fields are the basic stuff of physical (and all) reality, right?
If we take our current quantum field theory as a fundamental theory, yes. But nobody actually believes our current quantum field theory is a fundamental theory. So the only real answer is that we don't know what "the basic stuff of reality" actually is.
 
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  • #5
joneall said:
isn't the math supposed to represent physical entities?
The math helps us to make predictions about what happens in actual experiments. That does not mean everything in the math has to match up with something in reality. Nor does it mean that when something in the math does more or less match up with something in reality, that the something in reality has to be exactly the same as the thing in the math.

For example, as you say, quantum fields in the math are operators. Does that mean quantum fields in reality are operators? Of course not. That doesn't even make sense. Quantum fields in reality (assuming they are in reality--see the caveats in my previous post) are whatever they are; they don't have to be identical to the quantum fields in our math. All we need is for the math to make accurate predictions about what happens in reality.
 
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  • #6
joneall said:
Sorry, isn't the math supposed to represent physical entities? Is there a distinction I haven't grasped.
...and what are the "physical entities"?

(Imho) they're the various observables by which we measure and characterize all phenomena. In classical mechanics, observables are (physically) things like position, momentum, angular momentum, energy, etc. Mathematically, they are functions on phase space ##f(q,p)##, where ##q,p## denote position-like and momentum-like degrees of freedom respectively. Experiment tells us that any particular class of system can be characterized by a set of these observables satisfying a Lie algebra implemented by the Poisson bracket. (Alas, if you're not yet familiar with these, you'll first need to become proficient in Lagrangian and Hamiltonian mechanics before this will make much sense.)

Quantization basically amounts to representing the Lie algebra on Hilbert space instead, with a little help from the ##\hbar## constant. The familiar observables are just operators on this Hilbert space, constructed to satisfy (a slightly deformed version of) the classical Lie algebra. Just as classical observables ##f(q,p)## can be built up from the basic variables ##q,p##, quantum observables can be built from basic quantum fields, i.e., the operator equivalents ##\hat q## and ##\hat p##, defined to satisfy canonical commutation relations, analogous to how ##q,p## satisfy a canonical Poisson bracket equation.

[Hmm, I'm not sure if this is really an "I"-level answer but I guess we'll see...]
 
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  • #7
joneall said:
But, after second quantization, they are operators! I cannot get my head around this
Operators are functions, and it wouldn't really make sense to say that a function is something 'physical' or 'really there'. What is a quantum field 'really'? It's something that behaves in a way that closely matches the math that appears to govern it and has all the properties we associate with it. I can't think of another answer that would be functionally different in any significant way. When you boil things down this far, you really can't say anything else about things that science observes. After all, what is a dog? It's something that has all the behaviors and properties of what we call a dog. Same for anything else.
 
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  • #8
PeterDonis said:
But nobody actually believes our current quantum field theory is a fundamental theory.
Nobody? There are exceptions like me....
 
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  • #9
joneall said:
People ask me what quantum fields are. (Really...) Quantum fields are the basic stuff of physical (and all) reality, right? But, after second quantization, they are operators! I cannot get my head around this.
Why don't you start by asking them back, "What is a field?"
 
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  • #10
(Disregardign the issue that we don't know if current QFT needs future revision)
joneall said:
Sorry, isn't the math supposed to represent physical entities? Is there a distinction I haven't grasped.
One distinction I think is important is that the "representation" is physically encoded and operated by a context (which depending on your interpretation, is the observer, macroscopic environment, "human science", an agent, or some all universe average etc...).

In any of those cases, the math represents the physical entities in the "black box" only to the extent that it allows control and predictions from the given context. It represents what we(the context) know and can say about the "real entities" in the black box, given that we have incomplete initial information (which again can be explained differenly in different "interprerations"). And since anything we do know, must by inferred from interactions. What we _can know_ is also constrained, this is where the operators come into play.

I think it's hard to say more than this, without getting into a specific interpretation. About the relation between first and second quantization is also something that I think is hard to understand further than the math, unless one adopts a specific interpretation.

I am symphatetic to your question though, but you might need to find your own answer.

/Fredrik
 
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  • #11
I'm ok with basic Lie algebra. I'm wondering, though, whether the Ballentine strangerep referred to is Ballentine's book on QM or Ballentine's whiskey. The one may tend to lead to the other.
A. Neumaier said:
Nobody? There are exceptions like me....
I was going to say. Most of the textbooks I've seen leave you with this impression, mainly by not discussing the issue.
 
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  • #12
joneall said:
People ask me what quantum fields are. (Really...) Quantum fields are the basic stuff of physical (and all) reality, right? But, after second quantization, they are operators! I cannot get my head around this.
On the informal level, a quantum field is like a classical field, but with additional quantum fluctuations. These are negligible in the macroscopic domain, but dominate the submicroscopic domain since particles are excitations of these fluctuations - like little water wavelets but in an infinite-dimensional Hilbert space rather than in the 3-dimensional space of water waves.

On the mathematical level, quantum fields are described as fields of linear operators on Hilbert spaces, generalizing stress tensors, which are fields of linear operators on real 3-space (where linear operators are more familiar under the name of matrices).
 
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  • #13
Fra said:
In any of those cases, the math represents the physical entities in the "black box" only to the extent that it allows control and predictions from the given context. It represents what we(the context) know and can say about the "real entities" in the black box, given that we have incomplete initial information (which again can be explained differenly in different "interprerations"). And since anything we do know, must by inferred from interactions. What we _can know_ is also constrained, this is where the operators come into play.

I think it's hard to say more than this, without getting into a specific interpretation. About the relation between first and second quantization is also something that I think is hard to understand further than the math, unless one adopts a specific interpretation.
I'm wondering what these "interpretations" are that you are referring to.

When I posted this, I did not know what level to assign. I wanted basic down-to-earth answers, so gave it a "I" rating. I have been studying up on GR and QFT the last few years, after 35 years in computing. In my grad-school days (1962-67), there was no QFT course in my school. Lie groups were alluded to but never really explained.

I'm ok now with basic Lie algebra. (I'm wondering, though, whether the Ballentine strangerep referred to is Ballentine's book on QM or Ballentine's whiskey. The one may tend to lead to the other.)

Is Ballentine really good at explaining this sort of thing? Thanks for all your very helpful replies.
 
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  • #14
joneall said:
I'm wondering what these "interpretations" are that you are referring to.
https://en.wikipedia.org/wiki/Interpretations_of_quantum_mechanics ?

But as most interpretational questions are rooted in QM foundations, I don't recall it's often discussed them in a field context.

I personally have a "information processing theoretic" observer-centered interpretation of QM, which I use for fields just as single particles. It's just that a field containts more information with additional structure.

/Fredrik
 
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  • #15
A. Neumaier said:
Nobody? There are exceptions like me....
Even your thermal interpretation doesn't claim that QM is a fundamental theory, does it? There still have to be hidden stochastic degrees of freedom that QM does not represent, correct?
 
  • #16
martinbn said:
they are called by the same name.
Not just in English. In Italian, a field where you grow crops is a "campo", as is an electric field.

Actress Sally Field once said of her brother, physicist Rick Field, "He invented something called Field theory."
 
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  • #17
A field - and a quantum field is an example - is a mathematical object used to make accurate predictions of the physical universe. Is it "real"? I have no idea. That's metaphysics.
 
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  • #18
PeterDonis said:
Even your thermal interpretation doesn't claim that QM is a fundamental theory, does it?
No; this is not needed for theinterpretation to work.

But independent of the thermal interpretation, I believe that relativistic QFT augmented by canonically quantized gravity (with its infinitely many regularization parameters, which are probably an artifact of perturbation theory), and possibly another scalar field for dark matter, is the theory of everything.

By the way, It is very unlikely that I am the only physicist believing that QFT is really fundamental.
PeterDonis said:
There still have to be hidden stochastic degrees of freedom that QM does not represent, correct?
No. What are usually called hidden variables are in the thermal interpretation not hidden at all - they are the q-expectations, and they are deterministic, not stochastic. Stochasticity arises solely through the chaoticity of the deterministic dynamics, as everywhere in physics. Since we can observe only a small subsets of qauntities, this induces enough uncertainty to make the effective dynamics stochastic.
 
  • #19
Vanadium 50 said:
A field - and a quantum field is an example - is a mathematical object used to make accurate predictions of the physical universe. Is it "real"? I have no idea. That's metaphysics.
When teaching freshman classes about the concept of a (classical) field, I used to offer a generic "definition", sort of a field being any physical quantity that depended on position (at least). That worked, at least giving the class something concrete to work with.

(I always recall the remark of the 'wise guy' in a class, who asked whether time could be a field. I said "No", but then he said, "Time should be a field; otherwise why is my mother's age 50-something at home, but 40 at the hairdresser's?")
 
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  • #20
A. Neumaier said:
Stochasticity arises solely through the chaoticity of the deterministic dynamics
I see. Which means it comes from the fact that we cannot measure any q-expectations exactly, so we always have some ordinary statistical uncertainty about initial conditions.
 
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  • #21
Similarly to @Fra, I find it effective to think of QFT as a noisy signal analysis formalism. There has to be a data analysis component to that, which can be troublesome, but I will ignore that here. Signal analysis is about measurements, but it's realistic about real measurements never being point-like and always being distorted. We never see the world as it really is, we can only build models. We do what we can and we describe what we do as carefully as we can, but we don't complain about not being able to see the universe as it really is. In QFT, a "window function" describes how a given measurement of the signal is different from being point-like: it might be a small sphere, a large cube, or an oblate Gaussian weighted function; whatever it is, almost exactly the same role is played by a "smearing function" in QFT, often also called a "test function" (The difference is that a window function is used in convolution with the field, which convolves with the field we can't actually measure, so smearing is a little more basic, but it's the same idea.)
Instead of using ##\hat\phi(x)## for a measurement-operator-valued-distribution, aka a quantum field, use ##\hat M(x)##. A measurement operator in QFT (this is quantum mechanics, which is about measurements, right?) is then constructed as ##\hat M_f=\int\hat M(x)f(x)\mathrm{d}^4x##. If you look in older textbooks like Itzykson&Zuber, you'll find some discussion of this, but in path integral text books not so often.
With ##\hat M_f##, we can do some neat things. To begin, for a Gaussian vacuum state we can write down the smeared two-measurement vacuum expectation value as ##\langle v|\hat M_f\hat M_g|v\rangle=(f^*,g)##, where ##(f,g)## is a pre-inner product (so it can be zero even if ##f## and ##g## are non-zero). Then we can compute the probability density for the self-adjoint "##f##-measurement" ##\hat M_f^\dagger=\hat M_f## in the Gaussian vacuum state as the inverse fourier transform of $$\langle v|\exp(j\lambda\hat M_f)|v\rangle=\exp(-\lambda^2(f,f)/2),$$ which is a Gaussian with variance ##(f,f)##. We obtain $$\langle v|\delta(\hat M_f-u)|v\rangle=\frac{\exp(-u^2/(f,f)/2)}{\sqrt{2\pi(f,f)}}.$$ notice that it doesn't matter what the structure of the pre-inner product is, provided that the matrix ##(f_i,f_j)## is positive semi-definite, for whatever finite set of test functions we happen to be using, so this can be about ordinary QM as well as about QFT, but, I think, much more cleanly than you'll see most other places.
This lets us think of the vacuum state as a broadband, noisy carrier signal that we can modulate in various ways, as in this slide,
1700493863732.png

It's important to recognize that any modulated state is a global object that we only know anything about because we measure it, locally, in a way that is described by the ##f## in an ##f##-measurement. The state and measurements together is a higher-order mathematical object than just an ordinary classical field because we can modulate probability distributions. The vacuum state is definitely not like your local radio station, except that we can describe what the radio station does pretty well using only coherent modulations of the vacuum state. Note that the algebra lets us also compute what the probability distribution would be for any self-adjoint operator we can construct using ##\hat M_{f_1}, \hat M_{f_2}, ...##, but of course it all gets quite complicated.
I cover a lot of ground in that talk, but you can see the PDF on Dropbox for a little more about quantum fields in the slides surrounding the one I've copied in here. The PDF includes a link to the video on Syracuse University's website if anyone wants to really knock themselves out. What about interacting QFTs? Look at slide 22-... for a different story than I think you will find anywhere else.
All that said, I hope you find something that works for you. I don't discuss fermion fields at all, which is definitely a hit against me.
 
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  • #22
PeterDonis said:
I see. Which means it comes from the fact that we cannot measure any q-expectations exactly, so we always have some ordinary statistical uncertainty about initial conditions.
Yes. Since in QFT the q-expectations are the N-point functions, we'd need initial conditions for the N-point functions for all N=1,2,3,.... The dynamical equations are a hierarchy of equations coupling the N-point functions for different N. In the quantum case, the hierarchy is given by the Schwinger-Dyson equations; in the classical limit by the BBGKY hierarchy.

One can measure 1-point functions (smeared in space and time) and 2-point functions (via linear response theory) of Wightman fields. Examples of the latter are scalar fields, currents of spinor fields, and the electric and magnetic field.
Higher order N-point functions are very difficult to measure since they must be inferred indirectly from a huge number of observations (nonlinear response theory). 1-point or 2-point functions of the unobservable electromagnetic vector potential cannot be measured and are not beables in the thermal interpretation since the electromagnetic vector potential is not a Wightman field.

Even if we could measure 1-point and 2-point functions exactly we'd have some statistical uncertainty about the higher N-point functions, which are part of the dynamics. The hierarchy of equations must be truncated somehow; a number of useful numerical schemes are available, based on the fact (or assumption) that the truncated N-point functions seem to dwindle quickly with increasing N.
 
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  • #23
apostolosdt said:
(I always recall the remark of the 'wise guy' in a class, who asked whether time could be a field. I said "No", but then he said, "Time should be a field; otherwise why is my mother's age 50-something at home, but 40 at the hairdresser's?")
In Rovelli's relational approach to gravity, time is an observer-dependent field!
 
  • #24
PeterDonis said:
I see. Which means it comes from the fact that we cannot measure any q-expectations exactly, so we always have some ordinary statistical uncertainty about initial conditions.
It's obvious not consistent with measurements on single particles. We can measure a spin component of an Ag atom, prepared in a thermal state, with the Stern Gerlach experiment. The expectation value is ##\langle s_z \rangle=0##, but what's measured is either ##\sigma_z=+1/2## or ##\sigma_z=-1/2## with probabilities 50% for each outcome. The claim that we alwasy measure expectation values is at odds with observations as old as predating the discovery of modern QT!
 
  • #25
vanhees71 said:
It's obviously not consistent with measurements on single particles.
It is about measuring fields, not particles - cf. the subject of the thread.

In the thermal interpretation, particles are not beables; only their probability distributions are. For this is what objectively distinguishes particles prepared in different states; see your answer and my reply here.
 
  • #28
joneall said:
I'm wondering, though, whether the Ballentine strangerep referred to is Ballentine's book on QM or Ballentine's whiskey.
The word "copy" should have given you a clue. (Who would want a "copy" of Ballentine's whiskey??)

joneall said:
The one may tend to lead to the other.
Around 25-30 years ago I stopped keeping alcohol at home because I found that even 1 small bottle of cider at lunch interfered with my self-study of physics -- which needs every neuron of mental agility to be firing properly. So in fact, a concentrated persistent study of physics may have prevented me from becoming a shocking alcoholic like my father. :oldfrown:

joneall said:
Is Ballentine really good at explaining this sort of thing?
Well, clearly I think so, otherwise I wouldn't mention it in my signature space. (But note that his book is about QM, not full-on QFT.)
 
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  • #29
PeterDonis said:
The thermal interpretation's approach to Stern Gerlach measurements was discussed some time ago:

https://www.physicsforums.com/threa...-interpretation-explain-stern-gerlach.969296/
Yes, but this doesn't change the fact that we do not always measure the expectation value of an observable but in any outcome different values averaging to the expectation values when preparing a sufficiently large "statistical sample".
 
  • #30
vanhees71 said:
Yes, but this doesn't change the fact that we do not always measure the expectation value of an observable but in any outcome different values averaging to the expectation values when preparing a sufficiently large "statistical sample".
But this thread is about fields and not about particles.

When we measure a field we always measure some smeared q-expectation.
 
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  • #31
It depends on which observable you describe. E.g., if you deal with single unpolarized photons and measure the polarization, i.e., their helicity (to have a concrete "observable") you do not get the expectation value, which is of course 0 in this case, but either +1 or -1 (each with 50% probability). An expectation value represents a property of the ensemble/statistical sample not of the single member of the ensemble.
 
  • #32
vanhees71 said:
It depends on which observable you describe. E.g., if you deal with single unpolarized photons and measure the polarization,
In this thread we measure fields, not the polarization of a single particle.

The fields are the functions of E(x) and B(x) and include the polarization tensor, which is quadratic in these. Thus what we measure is a smeared 2-point function on a screen, which we get after sufficiently long exposure to the field in the beam, characterized by a thermal coherent state of the e/m field.
 
  • #33
A. Neumaier said:
In this thread we measure fields, not the polarization of a single particle.
vanhees71 does not think of photons as particles in your sense. A single particle here just means a single excitation of the photon field. I don't like the word particle in that sense either, but that is just how it is.

You also know some of those experiments were single photons or correlated pairs of photons are generated, and then measured, including correlations in their polarization.
 
  • #34
gentzen said:
vanhees71 does not think of photons as particles in your sense. A single particle here just means a single excitation of the photon field. I don't like the word particle in that sense either, but that is just how it is.
Photon particles are (both in the sense of vanhees71 and in my sense) very special states of the quantum field along a beam, namely sequences of 1-photon Fock states, or in the entangled case of 2-photon Fock states generated by parametric down-conversion, for use in Aspect type experiments).
gentzen said:
You also know some of those experiments were single photons or correlated pairs of photons are generated, and then measured, including correlations in their polarization.
Nevertheless a detector event at a photosensitive screen (or a pair of them in Aspect-type experiments) is not at all a measurement of the electromagnetic field at the screen(s). What would be the measured value of E(x) or B(x) at the screen?
 
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  • #35
Of course it is a measurement of the electromagnetic field. What else do you think you measure with a photoplate exposed by some electromagnetic radiation? You can nowadays prepare true single-photon states and observe the "detection-probability distribution" building up repeating the single-photon experiment again and again to build your "statical sample" (aka the "ensemble" in normal physics lingo).

It's also clear that a photon detector measures "smeared" correlation functions since it necessarily has a finite position resolution (e.g., finite extensions of pixels on a silicon detector) and also theoretically realistic observables are always smeared. Distributions/generalized functions do not represent "true states", as already well-known from introductory QM.
 

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