Physical interpretation of correlator

In summary, the two-point correlator is a quantity that can be interpreted physically as the probability amplitude for a particle to propagate from one space-time point to another. It is quantified by the sum of the amplitudes for different processes that give equivalent physical results.
  • #1
Frank Castle
580
23
Consider the 2-point correlator of a real scalar field ##\hat{\phi}(t,\mathbf{x})##, $$\langle\hat{\phi}(t,\mathbf{x})\hat{\phi}(t,\mathbf{y})\rangle$$ How does one interpret this quantity physically? Is it quantifying the probability amplitude for a particle to be created at space-time point ##(t,\mathbf{x})## and then propagate to, and be annihilated at space-time point ##(t,\mathbf{y})##?! Or does it quantify whether the field amplitudes at two different points are related or not (i.e. whether the values of the field operator at the two different space-time points are correlated or not)? Or is there another explanation?
 
Last edited:
Physics news on Phys.org
  • #2
It has no simple probabilistic interpretation. it is a vacuum expectation value of an operator.

The physical meaning is, like for all space or time correlators, in terms of linear response theory. This is treated in any book on nonequilibrium statistical mechanics. An example are the Green-Kubo relations.
 
Last edited:
  • Like
Likes vanhees71
  • #3
A. Neumaier said:
it is a vacuum expectation value of an operator.

How does one interpret the vacuum expectation value of a product of field operators though? In introductory texts that I've read in QFT, the quantity $$\langle\hat{\phi}(t,\mathbf{x})\hat{\phi}(t,\mathbf{y})\rangle := \langle 0\rvert T\lbrace\hat{\phi}(t,\mathbf{x})\hat{\phi}(t,\mathbf{y})\rbrace\lvert 0\rangle$$ is described as quantifying the probability amplitude for a particle to propagate from one space-time point to another. Wick's theorem can be used to express n-point correlators in terms of products of two-point correlators, so I assume that the interpretation of the two-point correlator can then be extended to interpretating the meaning of an n-point correlator?!)

Why are these vacuum expectation values of products of field operators (at different space-time points) referred to as correlators? The Wikipedia article states that "The correlation function can be interpreted physically as the amplitude for propagation of a particle or excitation between y and x." This seems a little vague - I assume that by amplitude the article is referring to probability amplitude, but I may be wrong here?!
 
Last edited:
  • #4
"Don't panic!" said:
Why are these vacuum expectation values of products of field operators (at different space-time points) referred to as correlators?
Because of the analogy to statistics where for Gaussian noise variables ##\eta_j## with zero mean and unit variances, ##\langle\eta_j\eta_k\rangle## is a correlation.
For free fields at equal time and with discrete space the analogy is quite close. For example, there is a ''Wick theorem'' for higher order expectation values of Gaussians long predating Wick 1948 (Isserlis 1918).
"Don't panic!" said:
The correlation function can be interpreted physically as the amplitude
Don't buy everything Wikipedia says. It cannot be a probability amplitude since unlike for the latter the absolute values can be arbitrarily large.
 
  • #5
A. Neumaier said:
For free fields at equal time and with discrete space the analogy is quite close

In this case is the correlation between the field values at two different spatial points?

A. Neumaier said:
Don't buy everything Wikipedia says. It cannot be a probability amplitude since unlike for the latter the absolute values can be arbitrarily large.

This leaves me confused though, as reading through Peskin and Schroeder, they give a similar interpretation. Zee's introduction to QFT also describe it as "the amplitude for a disturbance in the field to propagate from a point x to a point y".
I get what your saying, since the expectation value has a pole at ##p^{2}=m^{2}##, but it confuses me why such week known introductory texts describe it in this manner, which when first read seemed intuitively pleasing, but now feel disconcerted by.
 
  • #6
If it helps, Hatfield's QFT section 3.5 he gives the following probabilistic interpretation for a charged scalar field:

"Now consider the propagation of charge in the charge scalar field theory. The state corresponding to a particle of charge ##+1## at ##x## is ##\phi^*(x)|0>##, while the state corresponding to the particle at ##x'## is ##\phi^*(x')|0>##. Thus, the quantum mechanical amplitude to transport the charge from ##x## to ##x'## is $$<0|\phi(x')\phi^*(x)|0>.$$
Apparently, we can interpret the propagation of charge as the creation of a particle of ##+1## charge, an ##a## particle, out of the vacuum at ##x##, the transport from ##x## to ##x'##, and the reabsorption of the ##a## particle into the vacuum at ##x'##. Since we can't absorb the particle before it is created, this process only makes sense if ##t' \geq t##.

This is not the total amplitude for propagation of ##+1## charge. The total amplitude is the sum of all of the amplitudes of different processes that give equivalent physical results. In the process above, the charge at ##x## was increased by one unit, while the charge at ##x'## was lowered by one unit. We can accomplish the same thing by creating a particle of ##-1## charge, a ##b## particle, at ##x'## and transporting it to ##x##, then destroying it. Since ##\phi## creates ##b## particles and ##\phi^*## destroys them, the amplitude for this process is $$<0|\phi^*(x)\phi(x')|0>.$$
As before, we can't destroy a particle before it is created, so this process only makes sense if ##t \geq t'##.
The total amplitude, ##G(x',x)##, for propagation is the sum of the two amplitudes, $$G(x',x') = \theta(t'-t)<0|\phi(x')\phi^*(x)|0> + \theta(t-t')<0|\phi^*(x)\phi(x')|0>."$$
 
  • Like
Likes vanhees71
  • #7
"Don't panic!" said:
In this case is the correlation between the field values at two different spatial points?
yes.

"Don't panic!" said:
This leaves me confused though, as reading through Peskin and Schroeder, they give a similar interpretation. Zee's introduction to QFT also describe it as "the amplitude for a disturbance in the field to propagate from a point x to a point y".
I get what your saying, since the expectation value has a pole at ##p^{2}=m^{2}##, but it confuses me why such week known introductory texts describe it in this manner, which when first read seemed intuitively pleasing, but now feel disconcerted by.
Often ''amplitude'' is used just as a buzzword without a particular meaning, loosely suggesting an interpretation in Feynmans picture of particles traveling all possible paths.

This is a fictitious picture without any reality content. It serves to give a pseudo-intuition for abstract mathematical entities that helps beginners to swallow the formalism but later leads to stomach ache, as you experience it now.

As always in QFT, what counts is only the formulas, not the words, diagrams, and stories often associated with them. (see https://www.physicsforums.com/insights/misconceptions-virtual-particles/ for some loosely related stories.)
 
  • #8
bolbteppa said:
If it helps, Hatfield's QFT section 3.5 he gives the following probabilistic interpretation for a charged scalar field:

"Now consider the propagation of charge in the charge scalar field theory. The state corresponding to a particle of charge ##+1## at ##x## is ##\phi^*(x)|0>##, while the state corresponding to the particle at ##x'## is ##\phi^*(x')|0>##. Thus, the quantum mechanical amplitude to transport the charge from ##x## to ##x'## is $$<0|\phi(x')\phi^*(x)|0>.$$
Apparently, we can interpret the propagation of charge as the creation of a particle of ##+1## charge, an ##a## particle, out of the vacuum at ##x##, the transport from ##x## to ##x'##, and the reabsorption of the ##a## particle into the vacuum at ##x'##. Since we can't absorb the particle before it is created, this process only makes sense if ##t' \geq t##.

This is not the total amplitude for propagation of ##+1## charge. The total amplitude is the sum of all of the amplitudes of different processes that give equivalent physical results. In the process above, the charge at ##x## was increased by one unit, while the charge at ##x'## was lowered by one unit. We can accomplish the same thing by creating a particle of ##-1## charge, a ##b## particle, at ##x'## and transporting it to ##x##, then destroying it. Since ##\phi## creates ##b## particles and ##\phi^*## destroys them, the amplitude for this process is $$<0|\phi^*(x)\phi(x')|0>.$$
As before, we can't destroy a particle before it is created, so this process only makes sense if ##t \geq t'##.
The total amplitude, ##G(x',x)##, for propagation is the sum of the two amplitudes, $$G(x',x') = \theta(t'-t)<0|\phi(x')\phi^*(x)|0> + \theta(t-t')<0|\phi^*(x)\phi(x')|0>."$$

This is what I originally understood it as. I assumed that ##\langle 0\rvert T\lbrace\hat{\phi} (t,\mathbf{x}) \hat{\phi} (t',\mathbf{y}) \rbrace\lvert 0\rangle## describes a transition amplitude for a particle to propagate from a space-time point ##(t,\mathbf{x})## to the space-time point ##(t',\mathbf{y})##?! (Apologies, in my previous posts, I forgot to express the field operator at two different points in time as well as space).
 
  • #10
A. Neumaier said:
Often ''amplitude'' is used just as a buzzword without a particular meaning, loosely suggesting an interpretation in Feynmans picture of particles traveling all possible paths.

This is a fictitious picture without any reality content. It serves to give a pseudo-intuition for abstract mathematical entities that helps beginners to swallow the formalism but later leads to stomach ache, as you experience it now.

As always in QFT, what counts is only the formulas, not the words, diagrams, and stories often associated with them. (see https://www.physicsforums.com/insights/misconceptions-virtual-particles/ for some loosely related stories.)

I wish this wasn't done, it makes it much harder after one becomes a bit more familiar with the mathematics to actually understand the physics taking place.
How should one physically interpret the quantity ##\langle 0\rvert T\lbrace\hat{\phi} (t,\mathbf{x}) \hat{\phi} (t',\mathbf{y}) \rbrace\lvert 0\rangle## then?
 
  • #11
"Don't panic!" said:
How should one physically interpret the quantity ##\langle 0\rvert T\lbrace\hat{\phi} (t,\mathbf{x}) \hat{\phi} (t',\mathbf{y}) \rbrace\lvert 0\rangle## then?
All sorts of correlators are most naturally interpreted in a 4D Fourier transform with respect to the difference of all arguments, where they have a meaningful spectral (4D momentum) interpretation. To understand what they really mean you need to turn to nonequilibrium statistical mechanics.
 
  • #12
bolbteppa said:
we can interpret the propagation of charge as the creation of a particle
This only works when ##\phi## is a free field, and then it is clearly a fictitious process since free fields do not create anything.

In an interacting theory, ##\phi(x)|0\rangle## is not a single-particle state - particles have only asymptotic meaning.
 
  • #13
A. Neumaier said:
All sorts of correlators are most naturally interpreted in a 4D Fourier transform with respect to the difference of all arguments, where they have a meaningful spectral (4D momentum) interpretation. To understand what they really mean you need to turn to nonequilibrium statistical mechanics.

Would you be able to suggest any good texts on nonequilibrium statistical mechanics.

A. Neumaier said:
In an interacting theory, ϕ(x)|0⟩ϕ(x)|0⟩\phi(x)|0\rangle is not a single-particle state - particles have only asymptotic meaning.

In canonical quantisation QFT (in the interaction picture), doesn't the S-matrix element of a ##2\rightarrow 2## scattering $$\langle \mathbf{p}_{3},\mathbf{p}_{4}\rvert Te^{-i\int d^{4}x\;\mathcal{L}_{int}(x)}\lvert \mathbf{p}_{1}, \mathbf{p}_{2}\rangle$$ describe the scattering amplitude between asymptotically free particle states ##\lvert \mathbf{p}_{1}, \mathbf{p}_{2}\rangle## and ##\lvert \mathbf{p}_{3}, \mathbf{p}_{4}\rangle##, as ##t\rightarrow -\infty## and ##t\rightarrow\infty##, respectively.
Is this a pseudo-intuition as in the previous case we've been discussing?!
 
  • #14
"Don't panic!" said:
Would you be able to suggest any good texts on nonequilibrium statistical mechanics.
Initially Reichl, A modern course in statistical physics; this explains linear response theory and its relation to Green's functions and correlations in simpler, nonrelativistic classical and quantum situations. Later Calzetta and Hu, Nonequilibrium quantum field theory would be a good choice.
"Don't panic!" said:
In canonical quantisation QFT (in the interaction picture), doesn't the S-matrix element of a ##2\rightarrow 2## scattering $$\langle \mathbf{p}_{3},\mathbf{p}_{4}\rvert Te^{-i\int d^{4}x\;\mathcal{L}_{int}(x)}\lvert \mathbf{p}_{1}, \mathbf{p}_{2}\rangle$$ describe the scattering amplitude between asymptotically free particle states ##\lvert \mathbf{p}_{1}, \mathbf{p}_{2}\rangle## and ##\lvert \mathbf{p}_{3}, \mathbf{p}_{4}\rangle##, as ##t\rightarrow -\infty## and ##t\rightarrow\infty##, respectively.
"Don't panic!" said:
Is this a pseudo-intuition as in the previous case we've been discussing?!
S-matrix elements have a clear physical meaning. But one gets probability amplitudes only after integrating over a range of momenta corresponding to a normalized wave packet. The S-matrix elements themselves could perhaps be described to be ''probability density amplitudes''. But I think there is nothing gained by replacing the well-understood notion of an S-matrix element by such an expression.

Also, the time-ordered expression you wrote for the S-matrix makes sense only after renormalization (and only in the absence of bound states); so expanding it into free low order correlators doesn't give a good interpretation of free time-ordered correlators.

 
  • #15
A. Neumaier said:
Initially Reichl, A modern course in statistical physics; this explains linear response theory and its relation to Green's functions and correlations in simpler, nonrelativistic classical and quantum situations. Later Calzetta and Hu, Nonequilibrium quantum field theory would be a good choice.

Thanks, I'll take a look.

A. Neumaier said:
S-matrix elements have a clear physical meaning. But one gets probability amplitudes only after integrating over a range of momenta corresponding to a normalized wave packet. The S-matrix elements themselves could perhaps be described to be ''probability density amplitudes''. But I think there is nothing gained by replacing the well-understood notion of an S-matrix element by such an expression.

Also, the time-ordered expression you wrote for the S-matrix makes sense only after renormalization (and only in the absence of bound states); so expanding it into free low order correlators doesn't give a good interpretation of free time-ordered correlators.

Fair enough. Thanks for the information.
 

1. What is a correlator in physics?

A correlator in physics is a mathematical tool used to measure the degree of correlation between two quantities or variables. It is commonly used to analyze data and determine if there is a relationship between two physical phenomena.

2. How is a correlator used in experiments?

In experiments, a correlator is used to analyze data collected from measurements of two or more variables. By calculating the correlation between these variables, scientists can determine if they are related and if one variable can be used to predict the other.

3. What is the physical interpretation of a correlator?

The physical interpretation of a correlator depends on the specific variables being analyzed. In general, a positive correlator indicates a direct relationship between the variables, while a negative correlator indicates an inverse relationship. A correlator value of 0 indicates no correlation between the variables.

4. How does a correlator differ from a correlation coefficient?

A correlator and a correlation coefficient are similar in that they both measure the degree of correlation between two variables. However, a correlation coefficient is a single numerical value, while a correlator is a function that can take into account multiple variables and their relationships.

5. What are the limitations of using correlators in physics?

Correlators can only measure linear relationships between variables, so they are not suitable for analyzing non-linear relationships. Additionally, correlators can be sensitive to outliers in data, which can affect the accuracy of the results.

Similar threads

  • Quantum Interpretations and Foundations
Replies
5
Views
2K
  • Quantum Interpretations and Foundations
2
Replies
47
Views
1K
  • Quantum Interpretations and Foundations
2
Replies
54
Views
3K
  • Quantum Interpretations and Foundations
Replies
6
Views
1K
  • Quantum Interpretations and Foundations
Replies
2
Views
643
  • Quantum Physics
Replies
13
Views
948
  • Special and General Relativity
Replies
5
Views
266
  • Quantum Interpretations and Foundations
6
Replies
175
Views
6K
  • Quantum Interpretations and Foundations
Replies
2
Views
647
  • Quantum Interpretations and Foundations
9
Replies
309
Views
8K
Back
Top