# The stepwise path integral of the relativistic point particle

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## Abstract

In this paper we present a stepwise construction of the path integral over relativistic orbits in Euclidean spacetime. It is shown that the apparent problems of this path integral, like the breakdown of the naive Chapman–Kolmogorov relation, can be solved by a careful analysis of the overcounting associated with local and global symmetries. Based on this, the direct calculation of the quantum propagator of the relativistic point particle in the path integral formulation results from a simple and purely geometric construction.

## 1 Introduction

Local Lagrangian symmetries and relativity are essential in modern quantum physics. However, the simplest unification attempt fails dramatically: Lagrangian path integrals of relativistic point particles are considered intractable. The novelty of this article is that by considering a previously unnoticed symmetry, those problems are overcome and an exact calculation of the full propagator of the relativistic point particle is achieved.

If one remembers that the relativistic point particle is the simplest system with general covariance, it becomes clear that its understanding will be crucial for the consistent formulation of more complex theories with the same (extended) symmetry such as quantum gravity or string theory.

### 1.1 The problem

Since the first applications in non-relativistic quantum mechanics [1], the path integral formulation of quantum theories has developed in huge steps towards a quantum field theory of fundamental interactions. A common consequence of advancing in huge steps is that one leaves obstacles and possible subtleties unexplored on the way. This happened with the Path Integral (PI) formulation of the relativistic point particle. It is the purpose of this paper to close that gap and to resolve some misconceptions and problems that persisted until today in the context of this fundamental topic.

- 1Technical complications: At the first sight a technical issue arises from the appearance of non-Gaussian integrals. This issue can be avoided by the use of auxiliary field variables in the Hamiltonian action, which allow one to relate (at least at the classical level) the original action to a quadratic action [5, 6, 7], which is sometimes called einbein formalism. Those methods allow one to obtain the expected Klein–Gordon propagator from the PI of the relativistic point particle in
*D*dimensionsAs shown in Appendix A, a direct calculation of the PI with the Lagrangian action in$$\begin{aligned} K\sim \frac{1}{(k^2+M^2)}. \end{aligned}$$(1)*D*dimensions and without auxiliary fields is actually possible. The problem is, however, that it leads to a propagatorHere,$$\begin{aligned} K^{(n)}\sim \frac{1}{(k^2+M^2)^{n(D+1)/2}}. \end{aligned}$$(2)*n*is the number of intermediate slices. Apparently (2) does not have the expected form of (1). This difference was the first motivation for this study on the direct PI of the relativistic point particle. - 2Conceptual complications: Leaving the technical issues aside, there is a much more disturbing fact that complicates the understanding of the PI of the relativistic point particle: The Chapman–Kolmogorov (CK) equation for Markovian processes is not satisfied. This means that the standard notion of probability is not preserved in the process of free relativistic propagation. As an example, let us show this for the propagator (1) in position spacewhere \(\mathcal {N}\) is a normalization constant. The usual CK condition sates that: “Propagating from 0 to \(\varvec{x}_1\) and then from \(\varvec{x}_1\) to \(\varvec{x}_2\) and finally integrating over all \(\varvec{x}_1\), must be equivalent to propagating from 0 to \(\varvec{x}_2\)”. Applying this definition to the propagator (3) gives$$\begin{aligned} K(0,\varvec{x})=\mathcal {N}\int d^{D}k \frac{\exp (i\mathbf {k}\cdot \varvec{x})\sqrt{2}M}{k^2+M^2}, \end{aligned}$$(3)This is of course not the form of the original propagator (3), which is the well known and un-understood problem of the path integral of the relativistic point particle. In the literature there exist different stances on this embarrassing problem. Mostly, it is just taken as hint that at relativistic velocities the assumption of a single particle theory breaks down. It is argued that the energy available at such velocities would allow for interactions that again allow for the production of multi particle state [8]. This argument is, however, not very convincing since one was dealing with a free theory without interactions at the first place. Another stance is to try to fix this problem by the redefinition of the probability measure [11, 12].$$\begin{aligned} Kol(0,\varvec{x}_2)= & {} \int d^{D}x_1 K(0,\varvec{x}_1)K(\varvec{x}_1,\varvec{x}_2)\nonumber \\= & {} \mathcal {N}^2 \int d^{D}k \frac{\exp (i\mathbf {k}\cdot \varvec{x}_2)2M^2}{(k^2+M^2)^2}. \end{aligned}$$(4)

The paper is organized as follows: The introduction is completed by making notion of the symmetries of the relativistic point particle and by a definition of the PI measure taking into account those symmetries. In Sect. 2 several Euclidean PIs with one intermediate step are calculated in arbitrary dimensions and in Sect. 3 it is proven that those one-step propagators are already the full propagators for the given theory. Section 4 contains a discussion on the CK relation and the conclusion. Throughout the paper all formulas and discussions will be given in the imaginary time formalism corresponding to the Euclidean metric.

### 1.2 The relativistic point particle

*D*dimensions is

- (a)
Global Poincaré invariance: This can be seen from the fact that the action is invariant under global rotations and shifts of the coordinate system in

*D*dimensions. - (b)
Local Lorentz invariance: This means that the Lagrangian is invariant under local rotations in

*D*dimensions of the vector \((\mathrm{d} \mathbf {x})/(\mathrm{d}\lambda )\) at any point along the trajectory. A formal argument on why this symmetry, which is not a classical gauge symmetry, is actually important in this given context was given in [13]. - (c)
Weyl invariance: This means that the Lagrangian does not depend on the way that \(\lambda \) parametrizes a path \(\mathcal {P}\). The change to any other function \(\tilde{\lambda }(\lambda )\) would leave the Lagrangian invariant.

**a**) will be used to choose the coordinate system such that \(\mathbf {x}_i=0\) and that \(\mathbf {x}_f\) is different from zero in only one component. The symmetries (

**b**) and (

**c**) are symmetries which have to be treated with care when it comes to realizing an integral over different paths, since two seemingly different paths could be actually physically equivalent. The overcounting of physically equivalent paths would result in a wrong weight of some paths with respect to others.

### 1.3 General considerations on the explicit form of the PI measure

*t*, the number of time slicings

*n*used, and of the dimensions

*D*. Usually this normalization is fixed from imposing the Kolmogorov relations. However, there are several issues that arise when one tries to apply this naive measure (6) to the relativistic point particle which are all related to an overcounting of certain paths:

*n*and

*D*than the number of paths where this overcounting occurs. In simple words, it is very unlikely that \(\mathbf {x}_1\) happens to be on the classical path between \(\mathbf {x}_i\rightarrow \mathbf {x}_2\). However, in problems with local symmetries this might not necessarily be the case. In order to avoid overcounting of identical paths right from the start, one can improve (6) by stating for \(K=K(x_i,x_f)\)

Equation (9) is the definition of the measure \({\mathcal {D}}x\), which will now be used for the PI of the relativistic point particle.

## 2 Euclidean path integrals with one intermediate step

In this section we will discuss how the considerations on the measure, overcounting, and symmetries, are applied to the one slicing propagator \(K^{(1)}\). This will be explicitly done in one and two dimensions, before it is generalized to *D* dimensions. Before actually turning to the relativistic point particle it is instructive to discuss (9), in particular the meaning of the restriction \(|_\mathrm{NOC}\), for the case of non-relativistic quantum mechanics.

### 2.1 The non-relativistic path integral in two dimensions

This additional freedom corresponds to paths which are all redundant since they do not change any of the terms in (11). Thus, due to the \(|_\mathrm{NOC}\) condition in (9), those paths should not be included when integrating. The question that arises is: does this \(|_\mathrm{NOC}\) condition affect the usual propagator in non-relativistic quantum mechanics?

However, since all this symmetry fixing at the end of the day results in a simple rescaling of the normalization (14) by a factor of \(2\pi \), one sees that considering the symmetry (**b**) actually leaves all results of non-relativistic quantum mechanics unchanged. The situation is different for the relativistic path integral, as shown in the following subsections.

### 2.2 The relativistic path integral in one dimension

**c**) is already present in this case. As usual, one can fix this symmetry by choosing a unit length evolution parameter such that for each path

*M*for the one dimensional case

*n*intermediate steps, as will be discussed in a later section.

Remember that due to the fact that the one dimensional case does not have the local symmetry **b)** it was possible to waive all length dependent normalization factors defined in (9) and choose \({\mathcal {N}}({t, 1, 1})=1=\Delta _1\). For the two dimensional case, the symmetry **b)** is present and thus one has to consider those non-trivial contributions.

### 2.3 The relativistic path integral in two dimensions

However, the important feature of the relativistic case is that the Lagrangian (17) is the same for every single point along any of those paths with \(S=const.\) and \(\mathbf {x}_i,\, \mathbf {x}_f\) fixed. This is a reflection of the fact that local Lorentz symmetry (**b**) has not been fixed. Thus, if one naively counts all the points on the relativistic elliptic contour in Fig. 3, one is actually counting paths which are connected by a symmetry transformation. Instead, one should count only one point out of the elliptic contour by properly fixing the freedom introduced by the symmetry (**b**).

*x*axes and the vector \(\mathbf {x}_1-(\mathbf {x}_f-\mathbf {x}_i)\), as shown in Fig. 3. Further, for the Cartesian choice \(\mathbf {x}_i=0\), \(x_f=|\mathbf {x}_f- 0|\) stands for the length of the minimal classical path, while

*S*stands for the length of the quantum path \(0 \rightarrow \mathbf {x}_1 \rightarrow \mathbf {x}_f\). The Jacobian of the coordinate transformation is

By comparing the one slicing propagator \(K^{(1)}(x_f)=(2 \pi ) K_0(x_f M)\) with the zero slicing propagator \(K^{(0)}(x_f)\sim \exp (- M |x_f|)\), one notes that in contrast to the one dimensional case, the Kolmogorov relation is not fulfilled when going from zero to one slicing. This makes it necessary to study a higher number of intermediate steps, which will be done after discussing the one-step case in *D* dimensions.

### 2.4 The relativistic path integral in *D* dimensions

*D*dimensions \(\chi _i=(S,\alpha _1,...,\alpha _{D-1})\) one notes that for one slicing the Jacobian of the transformation takes the form

*D*dimensional spherical coordinates. Just like in the two dimensional case, the anomalous non-factorization of the angle \(\alpha _1\), can be corrected by an appropriate choice of \(\Delta _1\). An anomaly free symmetry fixing choice is again ensured for the definition (29), independent of the dimension

*D*. The remaining angular functions \(g(\alpha _1)\cdot f(\alpha _2, \dots \alpha _{D-1})\) do not mix with

*S*and give simply a solid angle. Further, in order to compensate the change in dimensionality induced by each additional spatial integral one has to choose the normalization with an inverse dimensional factor of \((|\mathbf {x}_f - 0|)\)

*S*one gets the one slicing propagator in

*D*dimensions

## 3 Euclidean path integrals with *n* intermediate steps

Up to now we have calculated the relativistic propagator with one intermediate step \(K^{(1)}_D\). However, according to (9) one still has to calculate infinitely many propagators with *n* intermediate steps \(K^{(n)}_D\) and than one has to sum them all up, taking again into account that no overcounting occurs. This sounds like a lot of work, unless one has some convenient relation (or theorem) that allows one to deduce all \(K^{(n)}_D\) and their sum just from the knowledge of \(K^{(1)}_D\). In non-relativistic quantum mechanics, this powerful tool of simplification is given in terms of the Kolmogorov relation (15). It will now be shown that there exists a generalization of this relation for the relativistic PI in one dimension and that for the relativistic PI in higher dimensions there exists a “nothing new theorem” which also allows one to deduce \(K^{(n)}_D\) from the knowledge of \(K^{(1)}_D\).

### 3.1 One dimensional case

*n*step propagator \(K^{(n)}(x_i, x_f)\) is also equal to (18) and that this propagator does fulfill the CK relation, as long as one avoids overcounting in the summation over intermediate steps \(x_1\). Thus, summing all \(K^{(n)}\) still gives

### 3.2 Two dimensional case

*n*intermediate steps

We now give a geometrical proof that the symmetry-fixed two-step propagator (38) is given by the one-step propagator (31). This proof is to be understood within the “No-Over-Counting” definition of the path integral measure (9).

- 1.
If \(\mathbf {x}_1\) is on the direct connection between \(\mathbf {x}_i\) and \(\mathbf {x}_2\), the step \(\mathbf {x}_1\) does not contribute a new path to (38) and those paths do not contribute to \(\int \mathrm{d}x_1^2 \int \mathrm{d}x_2^2 |_\mathrm{NOC}\).

- 2.
If \(\mathbf {x}_2\) is on the direct connection between \(\mathbf {x}_1\) and \(\mathbf {x}_f\), the step \(\mathbf {x}_2\) does not contribute a new path to (38) and those paths do not contribute to \(\int \mathrm{d}x_1^2 \int \mathrm{d}x_2^2 |_\mathrm{NOC}\).

- 3.
If none of the above cases applies, one knows that the path \(\mathbf {x}_1 \rightarrow \mathbf {x}_2 \rightarrow \mathbf {x}_f\) is equivalent (symmetry

**b**) to the path \(\mathbf {x}_1 \rightarrow \mathbf {x}_2' \rightarrow \mathbf {x}_f\) as indicated by the dashed lines in Fig. 4, where \(\mathbf {x}_2'\) is chosen such that \(\mathbf {x}_1\) lies on the direct line between \(\mathbf {x}_i\) and \(\mathbf {x}_2'\). As already shown, this different choice of \(\alpha _2\) does not change the measure contributed by this path. Thus, the total path \(\mathbf {x}_i \rightarrow \mathbf {x}_1 \rightarrow \mathbf {x}_2 \rightarrow \mathbf {x}_f\) is equivalent to the total path \(\mathbf {x}_i \rightarrow \mathbf {x}_1 \rightarrow \mathbf {x}_2' \rightarrow \mathbf {x}_f\). Since \(\mathbf {x}_1\) lies on the direct line between \(\mathbf {x}_i\) and \(\mathbf {x}_2'\), one falls back to scenario 1).

*n*steps and will always find that the propagator is given by (31). This also means that adding intermediate steps to the calculation of a propagator does not alter (31) and thus, the Kolmogorov relation holds trivially since there are no physically new intermediate points one can add. In ancient words: “There is nothing new under the sun”.

### 3.3 D dimensional case

Generalizing (35) to *n* intermediate steps in *D* dimensions is straightforward. Equation (41) holds also in this case, because one can set the angles \((\alpha _{i,1},...,\alpha _{i,N-1})\) such that new intermediate steps are on a straight line with the one intermediate step case. Thus, the “nothing new theorem” holds also in \(D\ge 2\) dimensions and the Kolmogorov relation for the relativistic point particle in *D* dimensions becomes trivial.

### 3.4 A check: Three dimensional non-relativistic PI with two intermediate steps

- 1.
If the point \(\mathbf {x}_2\) lies in the plane defined by \(\mathbf {x}_i, \mathbf {x}_1, \mathbf {x}_f\), in some cases a direct overcounting can happen. This occurs when \(\mathbf {x}_2\) lies on the continuation of the line \(\mathbf {x}_i \rightarrow \mathbf {x}_1\) (indicated by the red dashed line in Fig. 5), this configuration is actually an overcounting, since it is already considered by a one slicing path \(\mathbf {x}_i \rightarrow \mathbf {x}_2 \rightarrow \mathbf {x}_f\). This can happen, but it is actually numerically irrelevant since it corresponds to a one dimensional subset of the three dimensional volume \(d^3x_2\).

- 2.
The same argument holds if \(\mathbf {x}_1\) lies on the continuation of \(\mathbf {x}_f \rightarrow \mathbf {x}_2\).

- 3.
If the point \(\mathbf {x}_2\) lies outside of the plane defined by \(\mathbf {x}_i, \mathbf {x}_1, \mathbf {x}_f\) it still can happen that it is connected to an overcounting by a local rotation. Due to the rotational invariance explained in Fig. 2 one can always choose the arbitrary angle \(\alpha _2\) such that the transformed point \(\mathbf {x}_2'\) lies in the plane defined by \(\mathbf {x}_i, \mathbf {x}_1, \mathbf {x}_f\). If this point \(\mathbf {x}_2'\) lies on the continuation of \(\mathbf {x}_i \rightarrow \mathbf {x}_1\) one has an overcounting, otherwise not. All points \(\mathbf {x}_2\), which correspond to an overcounting, lie on a cone who’s tip is the point \(\mathbf {x}_1\), who’s symmetry axis is defined by the line \(\mathbf {x}_1 \rightarrow \mathbf {x}_f\), and who’s opening is given by the continuation of the line \(\mathbf {x}_i \rightarrow \mathbf {x}_1\). This configuration is shown by the green surface in Fig. 5. Again the volume of this overcounting is two dimensional which is negligible with respect to the three dimensional volume of \(d^3x_2\).

## 4 Discussion and conclusion

### 4.1 The Kolmogorov relation

It is interesting to discuss the results (35) and the D dimensional generalization of (41) in the context of the CK relation.

### 4.2 Conclusion

The usual path integral formulation of relativistic quantum mechanics suffers from deep technical and conceptual problems. A first problem appears when one uses the Lagrangian action and calculates the propagator in a straightforward way. It turns out that such a calculation of the propagator gives a wrong result (see Appendix A). Even though one can to some degree circumvent this particular problem by applying techniques such as renormalization procedures combined with semi-classical approximations [9], or the introduction of auxiliary fields for the Hamiltonian action [5, 6, 7], there remains a much deeper conceptual problem. This second problem is that it is not understood how the combination of short time propagators \(K^{(0)}\) gives long time propagators \(K^{(n)}\) of the same functional form, in the spirit of the Chapman–Kolmogorov relation. This failure is typically taken as hint that it is impossible to consistently formulate the PI of the relativistic point particle and that one has to turn to QFT instead (at least if one does not want to redefine the usual notion of probability [11, 12]).

Those two long standing issues are resolved in this paper. For this purpose, we make notice of the symmetries (**a**), (**b**), and (**c**) which are present in this problem. Then the usual PI measure is defined in a precise and explicit way (9), taking into account the issue of overcounting of equivalent and identical paths. Based on this, the relativistic one slicing propagator is calculated in a very simple and geometric way (35). The crucial step of the paper was then to show that it is indeed possible to consistently relate the n slicing propagator to the one slicing propagator by proving the relation (41). Thus, the one-step propagator (35) already gives the right result for the full propagator. This proof makes again heavy use of the symmetries and overcounting conditions discussed before. The main part of this paper concludes with a discussion on the Chapman–Kolmogorov relation and a conjecture on other quantum theories with general covariance.

It is hoped and believed that the presented work allows one to finally reconcile the quantum mechanical PI formulation with the straight forward notion of a relativistic action (5).

## Notes

### Acknowledgements

We want to thank I.A. Reyes and A. Faraggi for helpful discussions. B.K. was supported by Fondecyt No 1161150, E.M was supported by Fondecyt No 1141146.

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