**Motivation** How does one write down an action for the damped harmonic oscillator ? Denoting each time derivative as an overdot,

(1):

where is the mass, is friction, and is the oscillation frequency of the particle in the limit of zero friction.

More generally, how does one write down an action, not necessarily for particle mechanics, that does not require specifying boundary values?

It turns out that these questions are intimately related to the Schwinger-Keldysh formalism behind the computation of expectation values of quantum operators, as well as the treatment of out-of-equilibrium and/or open quantum systems. Here, I will merely focus on the (semi-)classical limit of two specific examples.

**General Strategy** The strategy goes as follows. First double the number of degrees of freedom. For example, if is the trajectory of the simple harmonic oscillator particle, we would now have and . The full action then takes the form

(2):

where the two are the same except one is evaluated on and the other on ; while the “influence action” couples the and but has to obey the anti-symmetry property:

(2′):

The difference between the action evaluated on copy-1 and that on copy-2 in eq. (2) arises, within the quantum context, from the Schwinger-Keldysh path integral when describing the time evolution of the density matrix, which plays a key role in the computation of expectation values of quantum field operators. Additionally, the influence action in eq. (2′) can be argued to arise from “integrating out” degrees of freedom.

The full action involves integrating the degrees of freedom from some initial time to the final time and — if fields (as opposed to particles) are involved — over the appropriate spatial domain. However, instead of the usual boundary values, one now requires that the copy-“1” and copy-“2” of the degrees of freedom to be specified at the initial time . At the final time we do not fix their trajectories but merely demand that the two copies coincide there:

(3):

This necessarily means their variation must also coincide:

(3′):

With these conditions in mind, we then demand that the total action be stationary under the variation of both copies of the degrees of freedom. Only after the ensuing equations-of-motion are obtained, do we set the two copies to be equal.

**Damped SHO** Let us now proceed to show that the damped harmonic oscillator of eq. (1) follows from the action

(4):

Demanding the action be stationary under variation with respect to both and ,

(5):

with the boundary terms

(5′):

Remember, from eq. (3), that we fix the initial conditions ; this sets to zero all the terms in the lower limit. Whereas, for the upper limit, we are to set ; ; as well as ; and only then does it vanish.

With the boundary terms vanishing, the principle of stationary action then yields the two independent equations

(5”):

and

(5”’):

Setting in equations (5”) and (5”’) then returns the DSHO equation of eq. (1).

**4D Majorana Fermion** For the second example, let us turn to the Majorana fermion, which unlike its Dirac cousin, only requires either the chiral left or chiral right SL spinor — but not both. One such version is provided by the equation

(6):

where is a 2-component spinor, is the identity matrix; with being the Hermitian Pauli matrices; is the fermion’s mass, and is the 2D Levi-Civita tensor. At the semi-classical level, and at first sight, you might think that the right hand side of eq. (6) could arise from a Lagrangian density of the form

(6′):

But upon closer examination you’d discover this Lagrangian density is identically zero*, as the Levi-Civita tensor is anti-symmetric and therefore

(6”):

But as it turns out, the doubled-field formalism allows one to write down a Lagrangian density. It is given by

(7):

where the Majorana mass term is now part of the `influence Lagrangian’ that couples the two copies:

(7′):

Notice the terms in eq. (7′) are similar to those in eq. (6′) but they do not vanish despite the anti-symmetric nature of , because we now have two distinct copies of the spinor field.

A similar variational calculation to the one performed for the DSHO would yield eq. (6) from the action in eq. (7). The primary difference from the DHO is that, fermionic systems are first order ones, and therefore only the two copies of the fields — but not their derivatives — need to match at , to ensure the boundary terms (analogous to the ones in eq. (5′)) vanishes.

I don’t yet know of any potential physical applications of such a perspective. What sort of open quantum systems would yield eq. (7′)?

* Upon quantization — as the referee of my paper below correctly emphasized — these Majorana fermion fields would still obey anti-commutation relations and, hence, Fermi-Dirac statistics. In fact, this is usually how the Majorana mass Lagrangian in eq. (6′) is justified: unlike the case of the Dirac mass terms, one has to introduce Grassmannian variables from the outset, so that eq. (6”) is no longer true .

References

- J. Schwinger, “Brownian Motion of a Quantum Oscillator,” J. Math. Phys. 2, 407 (1961).
- L. V. Keldysh, Zh. Eksp. Teor. Fiz. 47, 1515 (1964), [English translation, Sov. Phys. JEPT 20, 1018 (1965)].
- R. D. Jordan, “Effective field equations for expectation values,” Phys. Rev. D33, 444 (1986).
- C. R. Galley, D. Tsang and L. C. Stein, “The principle of stationary nonconservative action for classical mechanics and field theories,” arXiv:1412.3082 [math-ph].
- C. R. Galley, “Classical Mechanics of Nonconservative Systems,” Phys. Rev. Lett.
**110**, no. 17, 174301 (2013) doi:10.1103/PhysRevLett.110.174301 [arXiv:1210.2745 [gr-qc]]. - J. Polonyi, “Environment Induced Time Arrow,” arXiv:1206.5781 [hep-th].
- Y.-Z. Chu, “A Semi-Classical Schwinger-Keldysh Re-interpretation Of The 4D Majorana Fermion Mass Term,” J. High Energ. Phys., (2018) 2018: 13; arXiv:1708.00338 [hep-th].