What is Effective action: Definition and 21 Discussions

In quantum field theory, the effective action is a modified expression for the action, which takes into account quantum-mechanical corrections, in the following sense:
In classical mechanics, the equations of motion can be derived from the action by the principle of stationary action. This is not the case in quantum mechanics, where the amplitudes of all possible motions are added up in a path integral. However, if the action is replaced by the effective action, the equations of motion for the vacuum expectation values of the fields can be derived from the requirement that the effective action be stationary. For example, a field


{\displaystyle \phi }
with a potential


{\displaystyle V(\phi )}
, at a low temperature, will not settle in a local minimum of


{\displaystyle V(\phi )}
, but in a local minimum of the effective potential which can be read off from the effective action.
Furthermore, the effective action can be used instead of the action in the calculation of correlation functions, and then only one-particle-irreducible correlation functions should be taken into account.

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  1. J

    I Does the Dilaton field change particle mass?

    The low-energy effective action of the bosonic string is given by: $$S=\frac{1}{2k_0^2}\int d^{26}X\sqrt{-G}e^{-2\Phi}\Big(\mathcal{R}-\frac{1}{12}H_{\mu\nu\lambda}H^{\mu\nu\lambda}+4\partial_\mu\Phi\partial^\mu\Phi\Big)$$ where ##H_{\mu\nu\lambda}=\partial_\mu B_{\nu\lambda}+\partial_\nu...
  2. The black vegetable

    A Effective Action for Scalar and Fermion Fields

    I am reading Peskin and Schroeder Section 11.4. They derive a formula for the effective action p.372 Equation 11.63 using a scalar field interaction, They use this formula to determine the effective potential. If I want to do the same for a Lagrangian with with a scalar field and fermion...
  3. saadhusayn

    A Two loop Feynman diagram with quartic vertex

    I am trying to calculate the effective potential of two D0 branes scattering in Matrix theory and verify the coefficients in this paper: K. Becker and M. Becker, "A two-loop test of M(atrix) theory", Nucl. Phys. B 506 (1997) 48-60, arXiv:hep-th/9705091. The fields are expanded about a constant...
  4. saadhusayn

    Calculating effective action at two loops

    Does this mean that the expression for the above vertex is $$ -\frac{g}{2}\epsilon^{abx}\epsilon^{cdx}\int d\tau \langle A_{a} (\tau) A_{c} (\tau)\rangle \langle Y^{i}_{b} (\tau)Y^{i}_{d}(\tau) \rangle $$
  5. binbagsss

    A QHE ' the effective action should be a local functional'

    ' Finally, if we care only about long distances, the effective action should be a local functional, meaning that we can write is as ##S_{eff}[A]=\int d^d x...## ' Where does this come from and what does it mean? This isn't at all familiar with me, and I don't recall ever seeing anything...
  6. J

    A Does the effective action make sense in Quantum Mechanics?

    I think the effective action should make sense also in Quantum Mechanics, not only in QFT. But I have never seen described in a QM book as such. Could there be a QM book that uses effective actions? Or maybe in QM effective actions are called another name? I think effective actions in QM could...
  7. I

    Quantum QFT: groups, effective action, fiber bundles, anomalies, EFT

    Hi, I am looking for textbooks in QFT. I studied QFT using Peskin And Schroeder + two year master's degree QFT programme. I want to know about the next items: 1) Lorentz group and Lie group (precise adjectives, group representation and connection between fields and spins from the standpoint of...
  8. Ravendark

    Functional derivative of effective action

    1./2. Homework Statement In my QFT lecture we were introduced to the 1PI effective action ##\Gamma[\varphi]## for a scalar theory (in Euclidean space-time). In one-loop approximation we've found \Gamma^{(\text{1-loop})}[\varphi] = S[\varphi] + \frac{1}{2} \operatorname{Tr} \log D^{-1} where...
  9. S

    Low curvature effective action in string theory

    String effective action: S=-\frac{1}{2\lambda_{\text{s}}^{d-1}}\int d^{d+1}x\sqrt{|g|}e^{-\phi}\left[R+(\nabla\phi)^2+2\lambda_{\text{s}}^{d-1}V(\phi)-\frac{1}{12}H^2\right]+S_m where H^2=H_{\mu\nu\alpha}H^{\mu\nu\alpha}\\ H_{\mu\nu\alpha}=\partial_\mu B_{\nu\alpha}+\partial_\nu...
  10. W

    Question about Quantum Effective Action

    I am working on Quantum Effective Action in Weinberg QFT vol2 (page 67). In the last paragraph of page 67, the author said "Equivalently, ## i \Gamma [ \phi _0 ] ## for some fixed field ... with a shifted action ##I [ \phi + \phi_0 ]## : i \Gamma [ \phi _0 ] = ∫_{1PI, CONNECTED} ∏_{r,x}...
  11. K

    How to Derive the Chern-Simons Contribution to the Effective Action?

    Hi PF, I'm still very much a novice when it comes to QFT, but there's a particular calculation I'd like to understand and which (I suspect) may be just within reach. In short, the result is that after coupling a system of fermions to an external U(1) gauge field, one obtains a Chern-Simons...
  12. T

    Moments, Cumulants, and effective action?

    Hi, I'm studying quantum mechanics and statistical mechanics, and they make heavy use of the 'correlation functions/green's functions' which are merely the moments of the distribution of some variable. I have very intuitive understanding of moments and cumulants in terms of the distribution...
  13. M

    Currents and the quantum effective action

    Currents and the "quantum" effective action Hi all, I've been reading Burgess' Primer on effective field theory: arXiv:hep-th/0701053v2. I can't follow the reasoning here: I could maybe begin to make sense of this if I were allowed to assume that −(δ\Gamma/δϕ) is the sum of 1PI irreducible...
  14. N

    Is V_{eff} truly the density of effective action in QFT?

    Please teach me this: Why do we not call V_{eff}=\Gamma_{eff}/VT (where \Gamma_{eff} is the effective action, VT is spacetime volume) being density of effective action of field? Thank you very much in advance.
  15. Y

    QFT. Effective action and the skeleton expansion, how the legendre transform works

    QFT. Effective action and the skeleton expansion, how the legendre transform works! Homework Statement I've written a presentation on the effective action and have been posed a few questions to look out for. I think I know the answer to the first but am stumped by the second. "you've...
  16. P

    Effective action from the background field

    Hello, Assuming that I have a pure U(1) gauge theory. The partition function can be written as Z=\int D(A) \exp (-F_{\mu\nu} F^{\mu\nu}) If I want to find the effective action in terms of an external classical field I can write it in terms of A\rightarrow A+B where B is background and then...
  17. L

    Calculating the Effective Action of a Scalar Field Theory

    The effective action Γ[ϕ] for a scalar field theory is a functional of an auxiliary field ϕ(x). Both Γ and ϕ are defined in terms of the generating functional for connected graphs W[J] as W[J] + \Gamma[\phi] = \int d^dx J \phi , \quad \frac{\delta}{\delta J(x)} W[J] = \phi(x) Show - \int...
  18. J

    Supergravity as the low energy effective action of Superstring- / M-theory

    Hello. I have a question for the specialists on Superstring Theory: I am looking for a reference to original research, where the following is proven / shown / evidence is given / made plausible / etc. - 10D supergravities in their various forms are the low energy limits of superstring...
  19. B

    Computation of the effective action (Peskin 11.4)

    The generating functional Z[J] depends on \phi_{cl} trough its dependence on J. At the lowest order in perturbation theory the relation between J(x) and \phi_{cl} is just the classical field equation: \left(\frac{\delta L}{\delta \phi}\right)_{\phi=\phi_{cl}} + J(x) = 0 The question is...
  20. T

    Derivative Terms of Effective Action

    I know that the effective action can be written as a double expansion in derivatives and loop (h-bar). For example, take the effective action for a real scalar field: \Gamma[\phi]=\int...
  21. L

    Calculate Effective Action in Quantum Field Theory

    Can you tell me calculating method effective action in quantum field theory, please?