Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians

arXiv:2406.06530 · quant-ph · Submitted 2024-04-10 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians".

Mira: The gist: A new class of extended Lagrangians exists that are correlated to corresponding conventional Lagrangians without being homogeneous functions in the velocities, and with these extended Lagrangians,

Kai: First, who's behind it and why it matters.

Title and authors: Kai: So, we’re looking at a paper called "Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians." It’s by Jürgen Struckmeier from FIAS. Basically, they’re taking Feynman's path integral approach and putting it into a relativistic framework using this new kind of Lagrangian.

Mira: That sounds like it tries to fix a problem where you have space and time treated differently in the classical action, right? They’re introducing this extended Lagrange formalism where you treat physical time as a dependent variable on some evolution parameter s.

Mira: And when they project this non-homogeneous extended Lagrangian, which is called (twelve), it yields the standard conventional relativistic Lagrangian L q, dq dt, t = -mc squared s one - one/c squared dq dt squared + zeta c A dq dt - zeta phi > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg3,s 1 - 1/c^2 dq dt^2>.

The paper's summary: Kai: So, what we’re seeing is that they reformulate the classical dynamics by putting space and time on equal footing using these extended Lagrangians > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Mira: They do this by treating physical time as a dependent variable t(s) alongside the configuration variables q i(s), which lets them write the generalized action functional in terms of Le > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Lev: Essentially, they are using this new structure to rewrite the Euler-Lagrange equations so that they don't rely on time derivatives in the same way classical mechanics does, instead replacing them with d s terms > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Kai: The core idea is that they find a new class of extended Lagrangians that are correlated to the conventional ones without being homogeneous functions in the velocities > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Mira: This non-homogeneous structure is what lets them bridge the gap, and it’s what allows them to formulate a relativistic description that goes beyond just homogeneous forms > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Lev: The resulting set of equations, which they call equation (three), isn't redundant because it actually represents a holonomous constraint for the system’s evolution along s that separates classically unphysical states from those that are physical solutions > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg3>.

Kai: So, when you look at equation (three) or (eleven), it tells us something about how the system moves along s, which is what we need to understand the underlying physics > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg3>.

The paper's improvements: Mira: The paper suggests that one way to get this non-homogeneous extended Lagrangian (twelve) is by starting with a given conventional Lagrangian L and multiplying it by dt/ds > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg1,Relativistic generalization of Feynman's path integral on the basis of extended>.

Lev: And then they show that this leads to a trivial extended Lagrangian, which is one that contains no additional information about the underlying dynamical system because it’s homogeneous in the velocities > Relativistic Lagrangian L e = zero if L e not homogeneous zero if L e homogeneous > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg2>.

Kai: But they show that by considering a non-homogeneous extended Lagrangian, which is (twelve), we can determine in addition the correlation of the particle’s proper time with its laboratory time > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg1,Relativistic generalization of Feynman's path integral on the basis of extended>.

Mira: This correlation is a key physical insight because it means that L e goes beyond just being a formal construction and has real physical meaning to describe the same dynamics as that conventional Lorentz-invariant Lagrangian from (fourteen) > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg3,physical meaning to describe the same dynamics as>.

Lev: And this is what gives us the quadratic form of the velocity terms in L e, which is then lost when you project it into (fourteen), which makes (fourteen) unsuitable for a path integral formalism > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg3,the quadratic form of the velocity terms in>.

Kai: So, in short, they’re not just doing some formal math; they found a new class of Lagrangians that are physically significant because their quadratic velocity dependence is what lets them formulate a generalized path integral > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Conclusion: Mira: To wrap up, this paper shows that we can reformulate the classical dynamics using these extended Lagrangians to put space and time on equal footing > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Lev: The main implication is that we have a new class of Lagrangians for describing relativistic dynamics, especially those with quadratic velocity dependence which are useful for formulating a generalized path integral > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Kai: Because they show that the non-homogeneous extended Lagrangian (twelve) has physical meaning to describe the same dynamics as its conventional Lorentz-invariant Lagrangian from (fourteen) > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians <ref:2406.06530#pg3,physical meaning to describe the same dynamics as>.

Mira: This means we can build an AI that performs relativistic quantum simulations by employing this new method, and it yields the Klein-Gordon equation as the corresponding quantum description > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Lev: The main takeaway is that this paper provides a new tool for modeling particle dynamics in external fields, like electromagnetic fields, by utilizing these non-homogeneous extended Lagrangians > Relativistic generalization of Feynman's path integral on the basis of extended Lagrangians.

Jürgen Struckmeier

Frankfurt Institute for Advanced Studies

quant-ph

Submitted: 2024-04-10

Updated: 2026-10-14

Comments: 6 pages. arXiv admin note: substantial text overlap with arXiv:0811.0496

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 66/100

The gist: The gist: A new class of extended Lagrangians exists that are correlated to corresponding conventional Lagrangians without being homogeneous functions in the velocities, and with these extended

Key concepts

Extended Lagrange Formalism: This formalism reparametrizes classical dynamics by using an evolution parameter $s$ instead of physical time $t$. Physical time becomes dependent on $s$, allowing configuration variables and physical time to be treated symmetrically. The action is generalized from $L dt$ to a form involving the extended Lagrangian $ ext{Le} ds$, which describes the system's motion along this new parameter.
Non-homogeneous Extended Lagrangian ($ ext{Le}$): Unlike conventional Lagrangians that are homogeneous in velocities, this extended Lagrangian is not. It is quadratic in velocities and describes a particle's motion while simultaneously correlating its proper time with laboratory time. This form is crucial because it allows the path integral to be formulated correctly for relativistic quantum physics.
Proper-Time Path Integral: This approach adapts Feynman's path integral method to describe quantum mechanics using the extended Lagrangian. By approximating the action functional with an infinitesimal step $ ext{Le}_ ext{e}$, a transition amplitude between states is calculated. This procedure leads directly to the Klein-Gordon equation when imposing physical conditions derived from this formalism.
Klein-Gordon Equation Derivation: The non-homogeneous extended Lagrangian, when projected onto spacetime, yields a specific quantum mechanical equation. The resulting terms in the path integral expansion lead to an equation of motion that is mathematically equivalent to the Klein-Gordon equation for a charged spin-zero particle in an external electromagnetic field.

Terminology

Generalization — 2× in this paper · explained in 13 episodes
Quantum Physics — 2× in this paper · explained in 12 episodes

Summary

The gist: A new class of extended Lagrangians exists that are correlated to corresponding conventional Lagrangians without being homogeneous functions in the velocities, and with these extended Lagrangians, Feynman’s non-relativistic path integral approach can be converted into a form appropriate for relativistic quantum physics.

Formal Framework

The paper introduces the extended Lagrange formalism where the system’s dynamics is parametrized along a system evolution parameter s, and the physical time t is treated as a dependent variable t(s) on equal footing with all other configuration space variables q i (s) In the action principle, the conventional classical action L dt is then replaced by the generalized action Le ds Supposing that both Lagrangians describe the same physical system then provides the correlation of L and Le The extended Lagrangian Le is thus related to the conventional Lagrangian L by Le q, dq ds, t, dt ds = L q, dq dt, t dt ds, dq dt = dq/ds dt/ds.

Extended Action and Equations of Motion

The reformulation of the least action principle (1) that is eligible for relativistic physics is accomplished by treating the time t(s) = q 0 (s)/c — like the vector q(s) of configuration space variables — as a dependent variable of a newly introduced time-like independent variable, s [1–4]. The action functional then writes in terms of an extended Lagrangian Le Se[q(s), t(s)] = Z sb sa Le q, dq ds, t dt ds. The Euler-Lagrange equations that determine the particular path on which the value of the functional (2) takes on an extreme value adopt the customary form d s ∂Le/∂qµ ds − ∂Le/∂qµ = 0. For the index µ = 0, the Euler-Lagrange equation can be expressed equivalently in terms of t(s) as d s ∂Le/∂t ds − ∂Le/∂t = 0.

Correlation and Physical Significance

The actions, S and Se, are supposed to be alternative characterizations of the same underlying physical system. The extended Lagrangian Le is thus related to the conventional Lagrangian L by Le q, dq ds, t dt ds = L q, dq dt, t dt ds. For a non-homogeneous extended Lagrangian Le, the extended set of Euler-Lagrange equations (3) is not redundant. This equation (11) actually represents a holonomous constraint for the system’s evolution along s that separates classically unphysical states that do not satisfy Eq. (10) from those physical states that are solutions of the Euler-Lagrange equations (3).

Relativistic Quantum Description

The non-homogeneous extended Lagrangian (12) is furnished by the following Le that describes the motion of a relativistic point particle of mass m and charge ζ in an external electromagnetic field defined by the potentials (φ,A). The associated hypersurface condition (10) for Le coincides with that for the free-particle Lagrangian as all terms linear in the velocities drop out ds dt/ds squared = 1 ⇔ ds dt/ds = s 1 − 1/c squared dq dt squared. The non-homogeneous extended Lagrangian (12) thus determines in addition the correlation of the particle’s proper time with its laboratory time. With Eq. (13), the extended Lagrangian (12) may be projected into (T M)×R according to Eq. (5) to yield the well-known conventional relativistic Lagrangian L of the actual physical system L q, dq dt, t = -mc2 s 1 − 1/c squared dq dt squared + ζ c A dq dt − ζ φ. The rest energy term in Se,ǫ depends only on ǫ. It can, therefore, be taken as a factor in front of the integral and expanded up to first order in ǫ. This equation has the equivalent product form ∂/∂qα − iζ c Aα ∂/∂qα − iζ c A α ψ = mc/ 2 ψ, which constitutes exactly the Klein-Gordon equation for our metric ηµν.

Path Integral Derivation

For an infinitesimal step ǫ = sb - sa, we may approximate the action functional Se from Eq. (2) by Se,ǫ[qµ(s)] = ǫ Le qµb + qµa 2, qµb − qµa ǫ. The transition of a given wave function ψ(qµa) at the particle’s proper time sa to the wave function ψ(qµb) that is separated by an infinitesimal proper time interval ǫ = sb - sa can now be formulated as the path integral ψ(qµb) = 1/M Z exp i Se,ǫ ψ(qµa) d4 qa. The normalization factor M is now obvious. Since the equation must hold to zero order in ǫ, we directly conclude that M = (2π ǫ/im) 2. This means, furthermore, that the sum over all terms proportional to ǫ must vanish. The five terms that are linear in ǫ thus establish the equation m2 c squared 2 ψ = ∂ squared ψ/∂qα∂qα − ζ squared AαAα 2c squared ψ+ 2ζAα i c ∂ψ/∂qα + ζ i c ∂Aα/∂q α ψ.

Conclusion

Starting from the space-time formulation of the action principle, it was demonstrated that the Lagrangian description of classical dynamics can be reformulated in terms of extended Lagrangians in order to put space and time on equal footing. With the presentation of non-homogeneous extended Lagrangians that describe an unconstrained motion in an extended phase space, a new class of Lagrangians for the description of relativistic dynamics was found. Due to the quadratic velocity dependence of these Lagrangians, their usefulness to formulate a generalized path integral was shown. The non-homogeneous extended Lagrangian (12) thus determines in addition the correlation of the particle’s proper time with its laboratory time. With Eq. (13), the extended Lagrangian (12) may be projected into (T M)×R according to Eq. (5) to yield the well-known conventional relativistic Lagrangian L of the actual physical system L q, dq dt, t = -mc2 s 1 − 1/c squared dq dt squared + ζ c A dq dt − ζ φ. The quadratic form of the velocity terms in the Lagrangian (12) is lost owing to the projection, which renders the Lagrangian (14) unsuitable for the path integral formalism. We conclude that the non-homogeneous extended Lagrangian (12) is actually not a mere formal construction, but has the physical meaning to describe the same dynamics as the corresponding conventional Lorentz-invariant Lagrangian from Eq. (14). As the extended Lagrangian (12) is thus identified as physically significant, it can be concluded that the path integral erected on this Lagrangian yields the correct quantum description of a relativistic point particle in an external electromagnetic field.

References

[1] C. Lanczos, The Variational Principles of Mechanics (University of Toronto Press, Toronto, Ontario, 1949), Reprint 4th edn (Dover Publications, New York, 1986)

[2] J.R. Fanchi, Parametrized Relativistic Quantum Theory (Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993)

[3] F. Rohrlich, Ann. Phys. (N.Y.) 117, 292 (1979) <ref:2406.

Improvements for AI systems

The AI system can perform relativistic quantum simulations by employing a relativistic generalization of Feynman’s path integral using extended Lagrangians. This allows for the generalized path integral approach yields the Klein-Gordon equation as the corresponding quantum description.
The system can model particle dynamics in external fields, such as electromagnetic fields, by utilizing the non-homogeneous extended Lagrangian (12). This Lagrangian describes the same dynamics as the corresponding conventional Lorentz-invariant Lagrangian from Eq. (14), enabling more accurate physical modeling than purely formal constructions.
The AI can calculate wave function evolution for relativistic point particles in external fields by solving the path integral derived from the extended Lagrangian (15). This yields an analytical solution to Equation (20), which is equivalent to the Klein-Gordon equation for our metric ηµν.