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37

The main distinction you want to make is between the Green function and the kernel. (I prefer the terminology "Green function" without the 's. Imagine a different name, say, Feynman. People would definitely say the Feynman function, not the Feynman's function. But I digress...) Start with a differential operator, call it $L$. E.g., in the case of ...

29

The correlation function you wrote is a completely general correlation of two quantities, $$\langle f(X) g(Y)\rangle$$ You just use the symbol $x'$ for $Y$ and the symbol $x+x'$ for $X$. If the environment - the vacuum or the material - is translationally invariant, it means that its properties don't depend on overall translations. So if you change $X$ and ...

19

a very intuitive example for correlation functions can be seen in laser speckle metrology. If you shine light on a surface which is rough compared to the wavelength, the resulting reflected signal will be somehow random. This can also be stated as that you cannot say from one point of a signal how a neighbouring one looks like - they are uncorrelated. Such ...

17

This is just a property of Gaussian averaging analogous to the finite dimensional case: $\langle e^{ix} \rangle=\frac{1}{\sqrt{2\pi}\sigma}\int_{-\infty}^{\infty} e^{ix}e^{-\frac{x^2}{2\sigma^2}}=e^{-\frac{\sigma^2}{2}}= e^{-\frac{\langle x^2 \rangle}{2}}$ The field can be decomposed into its independent Gaussian modes and integrated for each mode ...

17

Excellent question, Kostya. Lubos already gave a detailed answer using general arguments in the language of QFT. In astrophysics and cosmology, however, there is another, and very simple, reason why we use the correlation functions all the time. It turns out that the mean value of the function $f(\vec{x})$, denoted $\langle f(\vec{x})\rangle$, can often not ...

10

For a given quantum system, the kernel of the path integral is, in fact, the kernel of an integral transform as you explicitly write down. It is the transform that governs time evolution of the system as is manifest in your first equation. For this reason, it is often referred to as the propagator of a given system. For example, for a single, ...

10

First, the term "propagator" is usually defined as the Green's function of the first type, not the second type, i.e. as a solution to the diffential equation $\hat L G = \delta$. At any rate, those definitions are ultimately equivalent – when the details are correctly written down – because the Green's function defined as the correlator in the second ...

9

The in- and out-states are free states, and the S-matrix definition of Mandl and Shaw is perfectly valid (with an appropriate notion of Texp). It is the one used in rigorous mathematical physics; see the treatise by Reed and Simon. It is also the one from which the LSZ formula is derived. It is the only way to define the S-matrix rigorously. The ...

9

In axiomatic approaches to quantum field theory, the basic field operators are usually realized as operator-valued distributions. That's what Wightman fields are: operator-valued distributions satisfying the Wightman Axioms. Wightman functions are the correlation functions of Wightman fields, nothing more. There's a nice theorem that says if you have a ...

8

@ArnoldNeumaier and @dushya have both pointed out correct solutions, but I want to elaborate a bit. The easy approach is the one dushya suggested. (You can also do what Arnold Neumaier suggests: First define the time-ordered product of operator-valued distributions, and then take expectation values.) Begin by recalling how Wightman functions are defined. ...

6

David Bar Moshe's derivation is of course right. Let me offer you a Taylor-expansion-based alternative proof: $$\left\langle e^{ix} \right \rangle = \left\langle \sum_{n=1}^\infty \frac{(ix)^n}{n!} \right \rangle = \left\langle \sum_{k=1}^\infty \frac{(ix)^{2k}}{(2k)!} \right \rangle$$ Here, I just used that by some odd-ness, the odd powers have a ...

5

For simplicity, let's restrict the discussion to that of a single particle moving in one dimension. Path integrals can be performed in much broader contexts like quantum field theory, but I think that would conceptually obscure the issue at this point. Let $H$ denote the (time-independent) quantum hamiltonian. Then the time evolution on the system is ...

5

I) Notational issues. First of all, be aware that Ref. 1 between eq. (4-27) and eq. (4-28) effectively introduces the retarded Greens function/propagator $$\tag{A} G(x_2,t_2;x_1,t_1)~=~\theta(\Delta t)~K(x_2,t_2;x_1,t_1), \qquad \Delta t~:=~t_2-t_1,$$ rather than the kernel/path integral $$K(x_2,t_2;x_1,t_1)~=~\langle x_2,t_2 | x_1,t_1 ... 5 The goal is to find the single particle propagator in the presence of interactions. This propogator will be the sum of all diagrams which have two external vertices. This sum of diagrams would be difficult to compute, but it turns out it easy to write this big sum of diagrams in terms of a sum of a smaller set of diagrams: the set of "one particle ... 5 Although there is already an accepted answer, I'll provide a different point of view here. It is important to keep in mind that the imaginary time is periodic, which means fields satisfy periodic(or anti-periodic) boundary conditions (the reason is because we use imaginary time path integral to represent the partition function, which is \mathrm{Tr}\, ... 4 Veneziano amplitude is a 4 tachyon amplitude in bosonic open string theory. Two tachyons are ingoing and two are outgoing. From the stringy point of view, tachyons are present in both closed and open bosonic theory and are the lowest particle in the spectrum, in particular they have negative mass squared: m_\textrm{open}^2=\frac{-1}{\alpha'}. In general a ... 4 |\Omega\rangle is the vacuum of the full interacting theory and |0\rangle is the vacuum of the free theory. They are related in the following way$$ |\Omega\rangle = \lim_{T\rightarrow(1-i\epsilon)\infty} (e^{-iE_0(T+t_0)}\langle \Omega|0\rangle)^{-1}e^{-iHT}|0\rangle $$and the correlators$$ \langle \Omega|\phi(x_1)\phi(x_2)...\phi(x_n)|\Omega\rangle = ...

4

Hint $\int_0^t \int_0^{t_1} dt_1 dt_2 \, a(t_1) a(t_2) = \frac{1}{2!} \int_0^t\int_0^t dt_1 dt_2\, \mathcal{T}\{ a(t_1) a(t_2) \}$ and so forth. You can see this by noting that the (square) integration region in the second integral can be split up into two triangular integration regions like in the first integral. This is one way to define the ...

4

The observables of the theory are, first of all, those in the algebra $\cal A$ (technically a $^*$-algebra with unit) of objects generated by the smeared fields $\phi(f)$. I mean linear combinations of $I$ and products of smeared fields $\phi(f)$, where $f$ is a complex valued compactly supported smooth function. This algebra can be enlarged by including ...

4

The primary utility in introducing the generating functional is in using it to compute correlation functions of the given quantum field theory. Let's restrict the discussion to that of a theory of a single, real scalar field on Minkowski space, and let $x_1, \dots, x_n$ denote spacetime points. Of central importance are time-ordered vacuum expectation ...

4

Right, one is only supposed to put the sources $J=0$ to zero after the very last $J$-differentiation has been performed. Figuratively speaking, short of writing out the calculation in full detail: Some of the $J$s downstairs can "couple" to the $J$s upstairs in the exponential.

4

Consider the case of a free scalar field, governed by the usual Lagrangian, $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2}m^2 \phi^2$$ The propagator, or equivalently Green's function for the theory is a function which can be though of as a response when we use a delta function as an input in the equations of motion, i.e. ...

4

Okay, so I gather from the link that $G^{(k)}$ in your notation refers to the correlation between field values at $2k$ points, with $\varepsilon^+$ inserted at half of them and $\varepsilon^-$ inserted at the other half. This concept of an $n$-point correlation function is very similar to the $n$th moment of a random variable or statistical distribution. ...

4

I think they do contribute to the S-matrix.The amplitude of disconnected diagrams is the product of the amplitudes of all disconnected pieces. For example, putting two connected 2-particle scattering diagrams will give you a 4-particle scattering process, but it's not that physically interesting because this process is not a "genuine" 4-particle process in ...

4

Your first answer is the correct one. The temporal auto-correlation function (or the time correlation function) of a fluctuating quantity $A(t)$ is $$C_{AA}(\tau) = \langle A(t) A(t+\tau)\rangle$$ This is a measure of how correlated $A(t)$ is to its value at another time $t+\tau$. The Green-Kubo formula that connects a response function (the electrical ...

3

$X$ isn't a Hermitian operator, as you've discerned, and $\langle 0 | X |0\rangle$ isn't the expectation value of an observable. It's traditional to call the quantum mechanical quantity $\langle 0| T(\phi(t_2) \phi(t_1) )| 0 \rangle$ a "correlation function", but this is actually an abuse of terminology. It is actually an amplitude! Or, if you prefer, ...

3


3

They are different. 'Schwinger terms' arise as central or abelian (or even non-abelian) extensions of current algebras of operators, very often as a consequence of the regularization procedure. They are source of anomalies. The original paper is short and readable: Field theory and commutators by J. Schwinger. You can read more in Moshe's answer to my ...

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