Εξίσωση Schwinger-Dyson

63.874pages on
this wiki

Εξίσωσις Schwinger-Dyson

- Μία Εξίσωση.

ΕτυμολογίαEdit

Η ονομασία "Εξίσωση Schwinger-Dyson" σχετίζεται ετυμολογικά με την λέξη " ".

ΕισαγωγήEdit

The Schwinger–Dyson equations (SDEs), also known as the Dyson–Schwinger equations, named after Julian Schwinger and Freeman Dyson, are general relations between Green functions in quantum field theories (QFTs). They are also referred to as the Euler–Lagrange equations of quantum field theories, since they are the equations of motion of the corresponding Green's function.

They form a set of infinitely many functional differential equations, all coupled to each other, sometimes referred to as the infinite tower of SDEs.

In his paper "The S-Matrix in Quantum electrodynamics",[1] Dyson derived relations between different S-matrix elements, or more specific "one-particle Green's functions", in quantum electrodynamics, by summing up infinitely many Feynman diagrams, thus working in a perturbative approach. Starting from his variational principle, Schwinger derived a set of equations for Green's functions non-perturbatively,[2] which generalize Dyson's equations to the Schwinger–Dyson equations for the Green functions of quantum field theories.

Today they provide a non-perturbative approach to quantum field theories and applications can be found in many fields of theoretical physics, such as solid-state physics and elementary particle physics.

Schwinger also derived an equation for the two-particle irreducible Green functions,[2] which is nowadays referred to as the inhomogeneous Bethe–Salpeter equation.

Derivation Edit

Given a polynomially bounded functional F over the field configurations, then, for any state vector (which is a solution of the QFT), $|\psi\rangle$, we have

$\left\langle\psi\left|\mathcal{T}\left\{\frac{\delta}{\delta\phi}F[\phi]\right\}\right|\psi\right\rangle = -i\left\langle\psi\left|\mathcal{T}\left\{F[\phi]\frac{\delta}{\delta\phi}S[\phi]\right\}\right|\psi\right\rangle$

where S is the action functional and $\mathcal{T}$ is the time ordering operation.

Equivalently, in the density state formulation, for any (valid) density state ρ, we have

$\rho\left(\mathcal{T}\left\{\frac{\delta}{\delta\phi}F[\phi]\right\}\right) = -i\rho\left(\mathcal{T}\left\{F[\phi]\frac{\delta}{\delta\phi}S[\phi]\right\}\right).$

This infinite set of equations can be used to solve for the correlation functions nonperturbatively.

To make the connection to diagrammatic techniques (like Feynman diagrams) clearer, it's often convenient to split the action S as S[φ]=1/2 D−1ij φi φj+Sint[φ] where the first term is the quadratic part and D−1 is an invertible symmetric (antisymmetric for fermions) covariant tensor of rank two in the deWitt notation whose inverse, D is called the bare propagator and Sint is the "interaction action". Then, we can rewrite the SD equations as

$\langle\psi|\mathcal{T}\{F \phi^j\}|\psi\rangle=\langle\psi|\mathcal{T}\{iF_{,i}D^{ij}-FS_{int,i}D^{ij}\}|\psi\rangle.$

If F is a functional of φ, then for an operator K, F[K] is defined to be the operator which substitutes K for φ. For example, if

$F[\phi]=\frac{\partial^{k_1}}{\partial x_1^{k_1}}\phi(x_1)\cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\phi(x_n)$

and G is a functional of J, then

$F\left[-i\frac{\delta}{\delta J}\right]G[J]=(-i)^n \frac{\partial^{k_1}}{\partial x_1^{k_1}}\frac{\delta}{\delta J(x_1)} \cdots \frac{\partial^{k_n}}{\partial x_n^{k_n}}\frac{\delta}{\delta J(x_n)} G[J].$

If we have an "analytic" (whatever that means for functionals) functional Z (called the generating functional) of J (called the source field) satisfying

$\frac{\delta^n Z}{\delta J(x_1) \cdots \delta J(x_n)}[0]=i^n Z[0] \langle\phi(x_1)\cdots \phi(x_n)\rangle$,

then, from the properties of the functional integrals

${\left \langle \frac{\delta \mathcal{S}}{\delta \phi(x)}\left[\phi \right]+J(x)\right\rangle}_J=0$,

the Schwinger–Dyson equation for the generating functional is

$\frac{\delta S}{\delta \phi(x)}\left[-i \frac{\delta}{\delta J}\right]Z[J]+J(x)Z[J]=0.$

If we expand this equation as a Taylor series about J = 0, we get the entire set of Schwinger–Dyson equations.

An example: φ4Edit

To give an example, suppose

$S[\phi]=\int d^dx \left (\frac{1}{2} \partial^\mu \phi(x) \partial_\mu \phi(x) -\frac{1}{2}m^2\phi(x)^2 -\frac{\lambda}{4!}\phi(x)^4\right )$

for a real field φ.

Then,

$\frac{\delta S}{\delta \phi(x)}=-\partial_\mu \partial^\mu \phi(x) -m^2 \phi(x) - \frac{\lambda}{3!}\phi(x)^3$.

The Schwinger–Dyson equation for this particular example is:

$i\partial_\mu \partial^\mu \frac{\delta}{\delta J(x)}Z[J]+im^2\frac{\delta}{\delta J(x)}Z[J]-\frac{i\lambda}{3!}\frac{\delta^3}{\delta J(x)^3}Z[J]+J(x)Z[J]=0$

Note that since

$\frac{\delta^3}{\delta J(x)^3}$

is not well-defined because

$\frac{\delta^3}{\delta J(x_1)\delta J(x_2) \delta J(x_3)}Z[J]$

is a distribution in

x1, x2 and x3,

this equation needs to be regularized!

In this example, the bare propagator, D is the Green's function for $-\partial^\mu \partial_\mu-m^2$ and so, the SD set of equation goes as

$\langle\psi|\mathcal{T}\{\phi(x_0)\phi(x_1)\}|\psi\rangle=iD(x_0,x_1)+\frac{\lambda}{3!}\int d^dx_2 D(x_0,x_2)\langle\psi|\mathcal{T}\{\phi(x_1)\phi(x_2)\phi(x_2)\phi(x_2)\}|\psi\rangle$
\begin{align} \langle\psi|\mathcal{T}\{\phi(x_0)\phi(x_1)\phi(x_2)\phi(x_3)\}|\psi\rangle =& iD(x_0,x_1)\langle\psi|\mathcal{T}\{\phi(x_2)\phi(x_3)\}|\psi\rangle + iD(x_0,x_2)\langle\psi|\mathcal{T}\{\phi(x_1)\phi(x_3)\}|\psi\rangle \\ &+ iD(x_0,x_3)\langle\psi|\mathcal{T}\{\phi(x_1)\phi(x_2)\}|\psi\rangle \\ &+ \frac{\lambda}{3!}\int d^dx_4D(x_0,x_4)\langle\psi|\mathcal{T}\{\phi(x_1)\phi(x_2)\phi(x_3)\phi(x_4)\phi(x_4)\phi(x_4)\}|\psi\rangle \end{align}

etc.

(unless there is spontaneous symmetry breaking, the odd correlation functions vanish)

There are not many books that treat the Schwinger–Dyson equations. Here are three standard references:

• Claude Itzykson, Jean-Bernard Zuber (1980). Quantum Field Theory. McGraw-Hill.
• R.J. Rivers (1990). Path Integral Methods in Quantum Field Theories. Cambridge University Press.
• V.P. Nair (2005). Quantum Field Theory A Modern Perspective. Springer.

There are some review article about applications of the Schwinger–Dyson equations with applications to special field of physics. For applications to Quantum Chromodynamics there are

ΥποσημειώσειςEdit

1. F. Dyson (1949). "The S Matrix in Quantum Electrodynamics". Phys. Rev. 75: 1736. doi:10.1103/PhysRev.75.1736. Bibcode1949PhRv...75.1736D.
2. 2,0 2,1 J. Schwinger (1951). "On Green's functions of quantized fields I + II". PNAS 37: 452–459. doi:10.1073/pnas.37.7.452. Bibcode1951PNAS...37..452S.

ΙστογραφίαEdit

Κίνδυνοι Χρήσης

Αν και θα βρείτε εξακριβωμένες πληροφορίες
σε αυτήν την εγκυκλοπαίδεια
ωστόσο, παρακαλούμε να λάβετε σοβαρά υπ' όψη ότι
η "Sciencepedia" δεν μπορεί να εγγυηθεί, από καμιά άποψη,
την εγκυρότητα των πληροφοριών που περιλαμβάνει.

"Οι πληροφορίες αυτές μπορεί πρόσφατα
να έχουν αλλοιωθεί, βανδαλισθεί ή μεταβληθεί από κάποιο άτομο,
η άποψη του οποίου δεν συνάδει με το "επίπεδο γνώσης"
του ιδιαίτερου γνωστικού τομέα που σας ενδιαφέρει."

Πρέπει να λάβετε υπ' όψη ότι
όλα τα άρθρα μπορεί να είναι ακριβή, γενικώς,
και για μακρά χρονική περίοδο,
αλλά να υποστούν κάποιο βανδαλισμό ή ακατάλληλη επεξεργασία,
ελάχιστο χρονικό διάστημα, πριν τα δείτε.

Επίσης,
Οι διάφοροι "Εξωτερικοί Σύνδεσμοι (Links)"
(όχι μόνον, της Sciencepedia
αλλά και κάθε διαδικτυακού ιστότοπου (ή αλλιώς site)),
αν και άκρως απαραίτητοι,
είναι αδύνατον να ελεγχθούν
(λόγω της ρευστής φύσης του Web),
και επομένως είναι ενδεχόμενο να οδηγήσουν
σε παραπλανητικό, κακόβουλο ή άσεμνο περιεχόμενο.
Ο αναγνώστης πρέπει να είναι
εξαιρετικά προσεκτικός όταν τους χρησιμοποιεί.

- Μην κάνετε χρήση του περιεχομένου της παρούσας εγκυκλοπαίδειας
αν διαφωνείτε με όσα αναγράφονται σε αυτήν

>>Διαμαρτυρία προς την wikia<<

- Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)