## Abstract

A review is made of the statistical generalization of microeconomics by Baaquie (Baaquie 2013 *Phys. A* 392, 4400–4416. (doi:10.1016/j.physa.2013.05.008)), where the market price of every traded commodity, at each instant of time, is considered to be an *independent random variable*. The dynamics of commodity market prices is given by the unequal time correlation function and is modelled by the Feynman path integral based on an *action functional*. The correlation functions of the model are defined using the path integral. The existence of the action functional for commodity prices that was *postulated* to exist in Baaquie (Baaquie 2013 *Phys. A* 392, 4400–4416. (doi:10.1016/j.physa.2013.05.008)) has been *empirically* ascertained in Baaquie *et al.* (Baaquie *et al.* 2015 *Phys. A* 428, 19–37. (doi:10.1016/j.physa.2015.02.030)). The model's action functionals for different commodities has been empirically determined and calibrated using the unequal time correlation functions of the market commodity prices using a perturbation expansion (Baaquie *et al.* 2015 *Phys. A* 428, 19–37. (doi:10.1016/j.physa.2015.02.030)). Nine commodities drawn from the energy, metal and grain sectors are empirically studied and their auto-correlation for up to 300 days is described by the model to an accuracy of *R*^{2}>0.90—using only six parameters.

## 1. Introduction

The field of microeconomics is primarily concerned with the prices of commodities and can also be viewed as a theory of commodity prices. This paper reviews the proposal made in [1] to formulate the theory of prices based on the concept of the action functional, and the subsequent publication [2] that provides strong empirical evidence in support of this formulation. The primary focus in the statistical microeconomics formulation is to describe the unequal time correlation functions of market prices. The auto-correlation function for single commodities is modelled using the action functional and Feynman path integral. The auto-correlation also provides a stringent test of the accuracy of the model.

The view taken in statistical microeconomics [1] is that supply and demand are inseparable and in fact are two facets of a single entity, namely the microeconomics *potential function* . The potential can be chosen to be the sum of supply and demand [1]
1.1
but can be more general, as discussed later.

The potential function , similar to mechanics, drives the evolution of market prices. For the special case when the prices are constant (time independent)—given by the constant prices **p**_{0}=(*p*_{01},*p*_{02},…,*p*_{0N})—the prices *minimize the value* of the potential; namely that is a minimum of .

The break-up of the microeconomic potential into a sum of supply and demand need not hold in general for all values of the price as the break-up is essentially an *asymptotic property* of the microeconomic potential. One expects from the behaviour of consumers and producers that the demand for a commodity increases with decreasing price and, concomitantly, the production of a commodity increases with increasing price. Hence, the most general microeconomic potential is stipulated to have the following two limiting cases:
1.2

In the framework of statistical microeconomics, stationary prices are determined by the minimum value of the microeconomic potential, which replaces the standard microeconomic procedure of setting supply equal to demand.

The microeconomic potential has its minimum value at its extrema , given by
What happens when ? The microeconomic potential then causes the prices to ‘move’, that is, to change and tend towards . Clearly, the more abrupt the change, the more unlikely it is; the change of price should, on average, be gradual and relatively ‘smooth’. To achieve this smooth movement the prices, in general, need a *kinetic term* .

In analogy with mechanics, the action functional is taken to be the sum of the potential term with the kinetic term , namely 1.3 The specific form of the kinetic term is determined by market data.

Although the concept of the kinetic term is taken from physics, it finds a natural expression in the evolution of the prices of commodities. It will be discussed later that is quite unlike the kinetic terms that appear in physics.

The action functional depends on the *function* **p**(*t*), : each possible function *p*(*t*) gives one numerical value for . For this reason is a functional of the price function and is called the action functional, or action in brief.

The Lagrangian is given by 1.4 The kinetic terms contain the time derivatives of the prices and, together with the potential function, determine the time dependence of the stochastic prices.

The dynamics of market prices is determined by assigning a *joint probability distribution* for all possible evolutions of the stochastic market prices. The probability distribution of the stochastic evolution of market prices is *postulated*, in analogy with quantum mechanics, to be the following:
1.5
The postulate encoded in equation (1.5) describes a classical random process that has a behaviour that is similar to that of a quantum mechanical system—with the expression determining the likelihood for the (different) random trajectories of the random prices.

## 2. Model of the microeconomic potential

The demand function is modelled to be [1]
2.1
and the supply function is modelled to be [1]
2.2
The coefficients *d*_{i},*s*_{i}, according to [3], are determined by macroeconomic factors such as interest rates, unemployment, inflation and so on. In the statistical microeconomics model, the coefficients *d*_{i}, *s*_{i} are determined from the historical prices of a commodity. It is our view that all the macroeconomic information that affects a commodity is contained in its price. Hence it is a consistency check to see if the values of *d*_{i}, *s*_{i} given by a macroeconomic analysis of a commodity agree with the result obtained by studying solely the price of a commodity.

The sum of the demand and supply function yields the microeconomic potential 2.3

The model microeconomic potential has the following expected asymptotic behaviour of exhibiting a supply and demand function as expected from equation (1.2):

Figure 1 shows the shape of for the model given in equation (2.3); note the important feature of —it has a (unique) global minimum at **p**_{0}. The value of **p**_{0} is obtained by *minimizing* and, from equations (2.3), yields the following:
2.4

In standard microeconomic theory, the market prices **p*** are fixed by equating demand to supply, shown graphically in figure 1; for the model being considered, this yields the following:

The market price of a commodity obtained from standard microeconomic theory (by equating supply to demand) yields a market price different from that obtained by minimizing the microeconomic potential. As one can see, except for the special case of *a*_{i}=*b*_{i}—which is *not* the case for the result obtained from the empirical study of the commodities—the market prices of the two approaches are quite different even for the same supply and demand functions.

The simple form of the microeconomic potential given in equation (2.3) allows one to write it as a sum of a demand and supply function for all values of the price. There can be complex cases of the microeconomic potential with multiple minima, as shown in figure 2, where the concept of a demand and supply function is only asymptotic, according to the general property of the microeconomic potential given in equation (1.2); for this potential, the absolute minimum of the potential fixes the market price and one cannot use the concept of supply equal to demand to fix the market price. In summary, the standard microeconomic theory of determining market prices by equating supply to demand is not valid in statistical microeconomics.

## 3. Microeconomic Lagrangian and action

The Lagrangian of a system determines the evolution of a dynamical system and for market prices represents all the factors determining its evolution. In particular, the interplay and competition of demand and supply with the ‘kinetic energy’ of market prices is encoded in the Lagrangian.

The Lagrangian, from equation (1.4), is given by the sum of the kinetic and potential factors and yields The action functional determines the dynamics (time evolution) of market prices and, from equation (1.3), is given by

Prices are always positive and hence are represented by exponential variables as *p*_{i}=*p*_{0} e^{xi}. The model chosen for the potential and kinetic parts of the Lagrangian yields, from equations (2.3) and (1.4), the following [1]:
3.1
and
3.2
The quantities are all real and independent parameters. Matrix *L*_{ij} is symmetric and positive definite, whereas matrix is symmetric but *not* positive definite. The Lagrangian given in equation (3.1) is nonlinear.

The Lagrangian for multiple commodities given in equation (3.1) has been proposed in [1]; a generalized form on equation (3.1), with the potential of the multiple commodities to have cross-terms of the form has been empirically studied in [4] for 18 major commodities.

For the case of a single commodity, let the price be *p*=*p*_{0} e^{x}; the Lagrangian given in equation (3.1) reduces to the following:
3.3

## 4. Market prices

The market price of a commodity, at time *t*, is denoted by
Commodity prices used in the empirical analysis are taken from the following website: http://www.investing.com/commodities/real-time-futures.

Standard microeconomic theory [5] usually assumes that, at time *t*, there is one unique market *p*(*t*) for a given commodity. We take the view that there is no unique price for a given commodity. Instead of a commodity having a unique price at each instant, the commodity price is assumed to be a random variable, having a range of prices, from zero to infinity; as the commodity price evolves in time, the commodity price undergoes a *continuous stochastic process*, with the price at any instant being inherently random and uncertain [1].

The *general equilibrium theory* of microeconomics studies the interplay of supply in an economy with multiple markets, and shows that an equilibrium is reached such that all prices are in equilibrium and take their market value [5].

In statistical microeconomics, all the prices are fundamentally *dynamical* and changing, as expressed by the kinetic terms in the Lagrangian given by

Hence, in contrast with the standard microeconomic theory, the dynamical nature of a stochastic price means that it is randomly changing and evolving in time. In particular, the prices are not the result of an equilibrium between supply and demand—as is expressed in the general equilibrium theory—but, instead, market prices are determined by the form of the action functional .

The microeconomic kinetic term is quite unlike the case for physics, for which only the ‘velocity’ term (∂*x*/∂*t*)^{2} appears. In fact, it is shown in [6] that the acceleration term (∂^{2}*x*/∂*t*^{2})^{2} is forbidden in quantum mechanics as it leads to a time evolution that does not conserve probability. As the prime purpose of modelling price is to explain the unequal time correlation functions, there is no need for a probabilistic interpretation of the action functional, as is the case for quantum mechanics.

The fundamental assumption of statistical microeconomics is that the behaviour of the price of a commodity is described by the microeconomic Lagrangian, a model of which is given in equation (3.3). The probability of the different prices being observed is determined by a probability density functional that is, up to a proportionality constant *Z*, given by the action functional as follows:

Prices are taken to be inherently random, and it is assumed that what one observes in the market are *samples* of the prices considered as a random variable. The observed time series of market prices is the result of sampling the random variations of the prices.

## 5. Microeconomic Feynman path integral

As prices follow a stochastic process, the appropriate description of prices is to determine the observed properties of the prices in terms of a statistical average over all possible values of the prices. The Feynman path integral is a mathematical formalism that provides an efficient procedure for evaluating these statistical averages. The mathematical aspects of the path integral are discussed in detail in [6] and reviewed in appendices A and B.

As discussed above in §4, the observed market prices are taken to be a random sample of the random price. The appropriate description of prices considered as a stochastic process is to calculate its various expectation values. The correlation function of prices provides a description of stochastic systems and in particular provides a measure of how the stochastic prices vary over time. The correlation function calculated from the model can be compared with the observed market value of these expectation values and hence provides a precise test of the accuracy of the model.

The correlation function of market prices is given by the expectation value of the product of prices, computed by summing over all possible histories of market prices using the path integral, and is given by the following: 5.1 where

The path integral measure is discussed in appendix B. The path integral consists of one integration—over all values of the price—for each instant of time; heuristically, this yields There are many techniques for giving a precise definition of [6].

One procedure is to discretize time into a lattice and consider only a finite number of lattice points; this truncation renders the path integral into an ordinary multiple dimensional integral—and is the basis of the numerical study of the path integral discussed in [2]. One has to take the limit of zero lattice spacing to obtain the continuous path integral.^{1}

The path integral for commodity prices given in equation (5.1) is nonlinear and nontrivial. The path integral can be studied perturbatively using Feynman diagrams and numerically using Monte Carlo and other well-known methods. In many cases, the numerical approach is necessary for studying features that are inaccessible to a perturbation expansion.

Figure 3 shows one sample value of the paths of the prices and the velocity and acceleration of these paths, namely *x*, ∂*x*/∂*t* and ∂^{2}*x*/∂*t*^{2}, over which the Feynman path integral is defined. The market values of velocity and acceleration ∂*x*/∂*t* and ∂^{2}*x*/∂*t*^{2} have been obtained by using finite differences.

### (a) Expansion of the microeconomic potential

Recall that, in equation (3.3), the log of price is defined by . Writing the potential in terms of variable *x* is appropriate for studying the action functional near its maximum as small variations of the price about the maximum are reflected in small variations of *x* [1]. The Lagrangian is given by
with
Note that the minimum of the potential equation (2.3), which is given in equation (2.4) and equal to , fixes the market price. Expanding the action functional about the minima of the potential yields the following:
where
5.2

### (b) Gaussian propagator

The unequal time correlation function of the log of two prices, called the propagator in physics, is given by
The Lagrangian has the form . We will show later that a more general quadratic term arises in the Lagrangian due to nonlinearities and we hence parametrize the quadratic Lagrangian by the following:
The Lagrangian yields the following exact propagator [7]:
The action is invariant under the shift of the time variable, and hence *G*(*t*,*t*′)=*G*(*t*−*t*′). This yields

Using a Fourier transform to evaluate the propagator for the prices yields with

*Case I*: ; *a*^{±} real, let ,

*Case II*: ; *a*^{±} complex
In summary, the complex branch yields the propagator
5.3

Empirical studies show that the behaviour of commodities is modelled by the complex branch, and hence our choice of domain and additional constraints are as discussed above. In fact, we shall later see that our model fits the market data quite well.

Note that the *market* correlation function additionally exhibits oscillations that do not vanish even for large *t*−*t*′, for which the model correlation function has decayed to zero. The value for *t*−*t*′ for which the model correlation function *G*(*t*−*t*′) is zero indicates the range for which the model is applicable, and which is an important consideration in fitting market data.

## 6. Calibrating the propagator

The prices and volatility of different commodities vary over a wide range and the action should be written in terms of variables that factor out the scale of the prices and their volatilities. With this in mind, we define a new set of variables *y*(*t*) by the following change of variables from *x*(*t*) to *y*(*t*):
6.1
where and *σ*(*x*) are the mean and standard deviation of the log of prices *x*(*t*). For the scaled variable *y*(*t*) that is of *O*(1), an action functional can be written that has the same form for different commodities.

The empirical values of the parameters *Γ*, *L* and are obtained in the following manner. The value of *G*(*t*−*t*′) is computed empirically for a range of time; the best fit of the model's value of *G*(*t*−*t*′) with its empirical value is obtained by varying the parameters of the model.

We define the propagator as the connected auto-correlator defined by

One needs to decide what is the minimum sample size *N* that accurately reflects the behaviour of the time series. For different groups of commodities, such as energy, metals and grains, there is a minimum sample size *N*. The auto-correlation for *N*=100 shows spurious behaviour for most commodities because the dataset is too small; for crude oil, a sample size of *N*=200 days is the minimum size of the data for having a reliable estimate as it follows the same trend as a larger sample size of *N*=800 days of data [2].

The goodness of fit of a set of points *y*(*x*) when compared with its fitted value *y*_{fit}(*x*) is given by
where is the mean of *y*(*x*).

In general, we choose a fixed size *N* for a given group of commodities. The auto-correlations for crude oil and copper are illustrated in figure 4*a*,*b* respectively; for crude oil and copper, the fit is excellent, with *R*^{2}=0.94 and *R*^{2}=0.96, respectively.

For crude oil, as shown in figure 5, the fit depends on the period of calendar time of the prices as well as on how large a time lag one is modelling. As shown in figure 5, the fit is good for 200 days.

The prices of commodities are stationary due to the choice of the Lagrangian given in equation (3.1) and obey 6.2 where time lag is defined by

Note that the fits are only valid for a *finite duration* of time, fixed by the *maximum* time lag—which is fixed by the value of the time lag |*t*−*t*′| for which the propagator goes to zero. The range of validity depends on the commodity. For example, from figure 5 we see that, for crude oil, the propagator goes to zero for a time lag of about 200 days, whereas for copper it is 400 days, as shown in figure 4*a*,*b*.

If one tries to fit the model past the time lag for which the correlation function is zero, the fit fails because prices may have large correlation out to very large lag time, as shown in figure 5. One needs to break up the time interval into sub-intervals of about 200 days and then obtain a good fit—with *R*^{2}>0.90—for each sub-interval by varying the parameters of the propagator.

Another possible limitation of the model is that for certain periods, when the market undergoes a sudden change, the fit may not be good. So far, for the all commodities that we have studied, the propagator always yields a good fit out to the first 200 days or longer, depending on the commodity.

The parameters for the best fits are given in the table 1 and are for the period 3 July 2013 to 21 January 2014; the fits for the nine commodities are all good, with *R*^{2}>0.91 for all of them.

## 7. Nonlinear terms: Feynman diagrams

The Gaussian propagator can only yield three parameters, namely and *Γ*, whereas the action has six parameters. Hence, we need to use the higher nonlinear terms in the action to fully calibrate the model.

The calibration of the nonlinear terms of the model is absolutely indispensable. The reason being that it is only the nonlinear terms that go beyond the Gaussian model and provide a microeconomic potential that has a minimum, and which can match the average market price of a given commodity. In particular, in the absence of the nonlinear terms, the quadratic potential yields all average market prices to be zero and is clearly quite useless for analysing market prices.

We show below that the values of the nonlinear terms are 10 times greater than the error terms, clearly showing that the value of the nonlinear terms is a defining feature of market prices. To check the consistency of the evaluation of the nonlinear terms using Feynman diagrams, a numerical simulation has been carried out in [2] to confirm that the range of the nonlinear terms obtained from the market data can in fact be obtained using the Feynman perturbation expansion.

The action is written in terms of the scaled variable *y*(*t*) and we obtain
where
and
Once we have obtained from market data, the potential parameters of *a*,*b*,*s*,*d* are then given by the following:

Expanding the action functional yields the following expansion: where Hence

The correlation function to leading order in the nonlinear coupling is shown in figure 6 and yields

For equal time, as *G*(*t*)=*G*(−*t*), we have
The ‘renormalized’ coefficient for the quadratic term is given by
7.1

To obtain the parameters *α*,*β*, we evaluate the expectation value of *y*^{3},*y*^{4}. The equal time *y*^{3} correlation is^{2}
7.2
The Feynman diagram for *E*[*y*^{3}]_{c} is shown in figure 7.

The equal time *y*^{4} correlation is given by^{3}
7.3
The Feynman diagram for *E*[*y*^{4}] is shown in figure 8.

## 8. The model's parameters for nine commodities

The empirical values of *α*,*β* for the different commodities are evaluated by comparing the analytical results for *E*[*y*^{3}] and *E*[*y*^{4}] given in equations (7.2) and (7.3) for *E*[*y*^{3}] and *E*[*y*^{4}] with the market data for nine commodities drawn from three different sectors.

We first fit the propagator that does not depend on *α*,*β*. We shift the value of *γ* to *Γ* as given in equation (7.1), and obtain the values of *L*, and *Γ*; table 1 gives the parameters that reproduce the market correlation function. We then obtain the values of *α* and *β* that reproduce the market values of *E*[*y*^{3}] and *E*[*y*^{4}], with the results given in table 2. The fact that the best fit for *α*, *β* lies within the perturbative range makes the calibration accurate and efficient.

A consistent interpretation of the potential function in terms of supply and demand requires that all the coefficients *a*,*b*,*c*,*d* are *positive*; the empirical values given in table 2 are all positive, hence verifying that the potential can be interpreted in terms of concept of supply and demand.

The complete calibration of the microeconomic Lagrangian for the different commodities allows us to express the potentials directly in terms of the prices of the commodities. Figure 9 shows the microeconomic potential of crude oil in terms of the three distinct variables, namely the scaled variable *y*, the logarithm of price *x* and, lastly, in terms of the price of the commodity *p*. Note that the volatility of the commodity plays an important role in determining the range of important fluctuations for the commodities price and the width of the microeconomic potential in figure 10, plotted against the price *p*=e^{x}*p*_{0}, reflects the value of the volatility of the various commodities. The microeconomic potentials for the other eight commodities are given in figure 10.

The empirical results obtained [2] show the key role of the kinetic term in the dynamics of commodity prices. Although both the kinetic and potential terms appear in the action functional, the market auto-correlation function for commodity prices shows the central role being played by the kinetic term; this term is absent in the standard treatments of microeconomics that are focused almost solely on supply and demand. A complete theoretical interpretation of the kinetic term in the framework of the theory of prices needs to be given and, as discussed earlier, the kinetic term is a reflection of the process of circulation and exchange of commodities as they transit from the producer to the consumer.

## 9. Conclusion

A model for microeconomics based on the action functional and path integral proposed by Baaquie [1] has been studied empirically in [2]. A collection of nine commodities from three different sectors were analysed to ascertain the validity and stability of the model.

The calibration and testing of the proposed statistical model of microeconomics are based on comparing the model's prediction with the empirical values of market prices. The model's propagator (unequal time correlation function) of market prices was obtained from market data in [1,3,7]. The Feynman perturbation expansion yields a consistent and efficient method for calibrating the nonlinear terms of the model.

The procedure adopted for the calibration of the model, and in particular obtaining the supply and demand functions, is based on the assumption that all the information about the behaviour of the commodities is contained in the observed market prices. This is quite unlike the procedure adopted in [3], where a large collection of macroeconomic data such as interest rates, unemployment, inflation and so on was required to estimate the supply and demand function. The approach adopted is also quite distinct from the auto-regression moving average (ARMA) [8].

In the statistical microeconomics approach [1], an *action functional* (as well as the Lagrangian)—consisting of the sum of a kinetic and a potential term—is *postulated* to exist in the market. The most important result of [2] is that an *action functional* does in fact *exist* in the market; the existence of the action functional for multiple commodities has been empirically verified in [4]. Hence, for both single and multiple commodities, the formulation of statistical microeconomics finds clear support from market prices. The microeconomic Lagrangian provides a self-contained and comprehensive framework for the study of microeconomics. In particular, one can now investigate what are the underlying theoretical principles of microeconomics that would give rise to an action functional formulation of statistical microeconomics.

## Competing interests

The author declares that he has no competing interests.

## Funding

We received no funding for this study.

## Acknowledgements

I thank Xin Du and Yu Miao for our collaboration and for many useful discussions. I also thank an anonymous referee for a critical reading of the manuscript that helped to clarify many points.

## Appendix A. The Feynman path integral

The main concepts of the Feynman path integral are reviewed as it forms the mathematical basis of the formalism of statistical microeconomics [6,9]. The discussion is based on the application of the path integral to physics and hence the form of the formalism has some significant differences from its application to microeconomics.

What is the probability amplitude if the quantum particle is only observed at its initial and final position? Owing to the quantum indeterminacy of a quantum entity, we expect that the entity's degree of freedom's path will be indeterminate and hence it will ‘take’ all possible *indeterminate paths* simultaneously.

How many indeterminate paths are there between the initial and final positions? Clearly, there are many paths, and to develop a sense of these paths, consider putting barriers between the initial and final position to *limit* the number of possible paths, as shown in figure 11, so that we can enumerate the indeterminate paths. Once the procedure for enumerating the indeterminate becomes clear, the barriers will be removed and all the indeterminate paths will then be included in our analysis.

Figure 11 shows a quantum particle going from initial state *x*_{i} at time *t*_{i} to a final position *x*_{f} at time *t*_{f}, through barriers that restrict the number of paths available to the quantum particle. Let the *entire continuous path*—going from initial state *x*_{i} to final state *x*_{f} through the successive slits as shown in figure 11—be denoted by path(*n*), with the probability amplitude denoted by *ϕ*[path(*n*)]. One can take path(*n*) to be straight lines from *x*_{i},*t*_{i} to the successive slit positions and another straight line from the last slit to *x*_{f},*t*_{f}, as shown in figure 11.

Consider the case where the particle is observed at initial time *t*_{i} to be at *x*_{i} and then another measurement is only performed at final time *t*_{f}—with the particle being detected at *x*_{f}. The barriers are placed between the initial and final positions, and let there be *N* total number of different paths going from *x*_{i} to *x*_{f}. There are *N* indeterminate paths from *x*_{i},*t*_{i} to *x*_{f},*t*_{f} that are all indistinguishable. From the superposition principle, the total probability amplitude is given by *adding* the probability amplitudes for all the indistinguishable determinate paths, and yields
A 1
The probability amplitude *ϕ*[path(*n*)] for *each determinate path* is given by the action functional and yields the following for the path(*x*_{i};*n*):
A 2
and
A 3
where is the action for path(*n*) and is a path-independent normalization.

Hence, from equations (9.1) and (9.2), the total space–time probability amplitude that the initial state vector |*x*_{i},*t*_{i}〉 makes a transition to the final state vector |*x*_{f},*t*_{f}〉—via *trans-empirical paths*—is given by superposing the amplitude for all the trans-empirical paths and yields

The evolution kernel (total transition amplitude) has the following representation: A 4

One can successively remove the barriers between the initial and final positions of the quantum particle, as shown in figure 11, and there will be great proliferation of possible paths. When there are no longer any slits, one has the limit of or what is the same thing, there are infinitely many trans-empirical paths.

The transition amplitude is given by the sum over *all possible trans-empirical paths*, going from the initial position *x*_{i} at time *t*_{i} to the final state *x*_{f} at time *t*_{f}, as shown in figure 12, and yields the following:
A 5
The sum in equation (9.5) looks more figurative than a precise mathematical expression. After all, how are we supposed to actually perform a sum over infinitely many paths? Equation (9.5) is recast into a precise and mathematical expression in appendix B using the techniques of quantum mechanics.

In summary, all the paths going from initial state *x*_{i} to final state *x*_{f} are *indistinguishable and hence trans-empirical* as no measurement is performed for the duration, from time *t*_{i} to time *t*_{f}, over which the system evolves. The total probability amplitude to make a transition from initial state *x*_{i} to final state *x*_{f} is equal to the sum over all the individual probability amplitudes to go from initial state *x*_{i} to final state *x*_{f}.

For physical (Minkowski) time, Schrödinger's equation yields
where *H* is the Hamiltonian operator. Minkowski time is analytically continued to Euclidean time *τ*, defined by
A 6
Propagation in Euclidean time is effected by operator *T*=e^{−τH} and yields
The reason for studying quantum systems in Euclidean time is to have a well-defined operator that is convergent and not oscillatory. In Minkowski time, one is faced with a similar oscillatory expression such as , which needs to be defined using the theory of distributions when .

Analytic continuation to Euclidean time entails *no loss of information* because *T* and *H* have the same eigenfunctions, with the Euclidean Hamiltonian equal to the original Minkowski Hamiltonian given by
The eigenvalues *T*_{n},*E*_{n} are related by
The path integrals that appear in microeconomics and quantum finance are formulated for describing systems that have classical randomness and are defined for Euclidean time.

## Appendix B. Path integral for evolution kernel

The evolution kernel (transition amplitude) is defined by

To render the sum over all paths in continuous space, namely and given in equation (9.5), into a well-defined mathematical quantity, a derivation is given of the path integral starting from the Schrödinger equation. A corollary result will be to show that the definition given of *K*(*x*_{i},*x*_{f};*t*) is equivalent to the one derived in equation (9.5).

The evolution kernel is evaluated in Euclidean time because the expressions are mathematically more rigorous and transparent than in Minkowski time. The Euclidean (imaginary time) evolution kernel is given by B 1

Note , and it is this non-commutativity that poses the main problem in quantum mechanics. Ignoring the non-commutativity yields e^{−τH}≃*e*^{−τ(p2/2m)} e^{−τV}, and for this case
B 2
and the evolution kernel *K* requires the kernel for the free particle Hamiltonian *p*^{2}/2*m*.

Note the remarkable fact that for non-commuting operators *A* and *B*
B 3
For *τ*=*ϵ*, infinitesimal time, one has the following result:
B 4
Hence for infinitesimal time *ϵ*, from equation (9.8) the transition amplitude *K*(*x*,*x*^{′};*ϵ*) can be evaluated exactly to *O*(*ϵ*^{2}).

The path integral approach is employed fundamentally to build up the finite time transition amplitude by composing the infinitesimal time transition amplitude by repeatedly using the resolution of the identity operator.

The evolution kernel (transition amplitude) for a particle to go from initial position *x*_{i} to final position *x*_{f} in time *τ* can be written as follows:
B 5
where, for *ϵ*=*τ*/*N*, we have
B 6
Inserting, *N*−1 times, the following completeness equation
yields the following:
B 7

Consider the matrix element
B 8
As 〈*x*|*p*〉=e^{ipx}, one has from equation (9.10)
B 9

The Dirac–Feynman formula, for each infinitesimal determinate step *ϵ*, is given by
The evolution kernel (transition amplitude) is defined by
Simplifying the notation yields the Lagrangian, defined for infinitesimal *Euclidean time* *ϵ*, given by
B 10
Hence from equation (9.15)
B 11

For the particle degree of freedom the Hamiltonian, with potential *V* (*x*), is given by
and its Lagrangian is given by equation (9.17), for discrete time *t*=*nϵ*, by
B 12
The Lagrangian is sometimes written more symmetrically as
B 13
and to *O*(*ϵ*) is the same as the one given in equation (9.18).

In summary, the transition amplitude is given by where the lattice action and path integral integration measure is given by B 14

In the continuum limit one obtains the following: B 15 The continuum Euclidean path integral representation for the evolution kernel is B 16

All paths between the initial and final position, figuratively shown in figure 12, are summed over the path integration given in equation (9.22). The figurative summation over all paths in Minkowski time given in equation (9.5) is given a mathematical realization in equation (9.22), which is an integration over all paths in Euclidean time.

At each instant, the position degree of freedom takes all its values; at instant *t*, the degree of freedom is equal to the real line ℜ_{t}; the total space of all paths is given by a tensor product over all instants and yields the total space of all paths equal to ⊗_{t}ℜ_{t}. In general, for degree of freedom space given by , the path space is given by .

In summary, the *Feynman path integral* is given by
The formulation of statistical microeconomics given in §5 is a specific example of the path integral. For systems that are described by a Hamiltonian operator given by *H* the path integral is an efficient mathematical instrument for evaluating the finite time matrix elements of the operator , namely of 〈*x*_{f}|e^{−τH}|*x*_{i}〉.

## Footnotes

One contribution of 14 to a theme issue ‘Quantum probability and the mathematical modelling of decision making’.

↵1 For a finite time path integral, discretizing time, as given in equation (9.20), yields

↵2

*E*[*y*^{3}(*t*)]_{c}=*E*[*y*(*t*)^{3}]−2*E*[*y*(*t*)]*E*[*y*^{2}(*t*)]=*E*[*y*(*t*)^{3}].↵3

*E*[*y*^{4}(*t*)]_{c}=*E*[*y*(*t*)^{4}]−3*E*[*y*^{2}(*t*)]*E*[*y*^{2}(*t*)].

- Accepted September 4, 2015.

- © 2015 The Author(s)