Probability Thermodynamics and Probability Quantum Field
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] A p r Probability Thermodynamics and ProbabilityQuantum Field
Ping Zhang, a,d ∗ Wen-Du Li, b,d
Tong Liu, c,d and Wu-Sheng Dai d ∗∗ a School of Finance, Capital University of Economics and Business, Beijing 100070, P. R. China b Theoretical Physics Division, Chern Institute of Mathematics, Nankai University, Tianjin, 300071,P. R. China c School of Architecture, Tianjin University, Tianjin 300072, P. R. China d Department of Physics, Tianjin University, Tianjin 300350, P.R. China
Abstract:
In this paper, we introduce probability thermodynamics and probability quan-tum fields. By probability we mean that there is an unknown operator, physical or non-physical, whose eigenvalues obey a certain statistical distribution. Eigenvalue spectra definespectral functions. Various thermodynamic quantities in thermodynamics and effective ac-tions in quantum field theory are all spectral functions. In the scheme, eigenvalues obey aprobability distribution, so a probability distribution determines a family of spectral func-tions in thermodynamics and in quantum field theory. This leads to probability thermody-namics and probability quantum fields determined by a probability distribution. There aretwo types of spectra: lower bounded spectra, corresponding to the probability distributionwith nonnegative random variables, and the lower unbounded spectra, corresponding toprobability distributions with negative random variables. For lower unbounded spectra,we use the generalized definition of spectral functions. In some cases, we encounter diver-gences. We remove the divergence by a renormalization procedure. Moreover, in virtue ofspectral theory in physics, we generalize some concepts in probability theory. For example,the moment generating function in probability theory does not always exist. We redefinethe moment generating function as the generalized heat kernel, which makes the conceptdefinable when the definition in probability theory fails. As examples, we construct exam-ples corresponding to some probability distributions. Thermodynamic quantities, vacuumamplitudes, one-loop effective actions, and vacuum energies for various probability distri-butions are presented. [email protected], [email protected]. ontents t -distribution 326.2.4 Laplace distribution 346.2.5 Cauchy distribution 356.2.6 q -Gaussian distribution 376.2.7 Intermediate distribution 40 A physical system is described by an operator. The eigenvalue of the operator containsonly partial information of the system. This is the reason why one cannot hear the shapeof a drum [1]. Nevertheless, the information embedded in eigenvalues, though partial, isof special importance. In physics, for example, in thermodynamics various thermodynamicquantities are determined by eigenvalues, and in quantum field theory the vacuum ampli-tude, the effective action, and the vacuum energy are determined by eigenvalues [2]. Inmathematics, a famous example of spectral geometry is given by Kac that one can extracttopological information, the Euler characteristics, from eigenvalues through the heat kernelor the spectral counting function [1, 3]. Quantities, such as thermodynamic quantities,effective actions, and spectral counting functions, are all spectral functions. The spectralfunction is determined by a spectrum { λ n } and, on the other hand, all information of thespectrum is embedded in the spectral function [4]. Once an eigenvalue spectrum is given,even if the operator is unknown, we can also construct various spectral functions.In the scheme, without knowing the operator, we preset an eigenvalue spectrum toconstruct spectral functions. The eigenvalue spectrum { λ n } is supposed to obey a certainprobability distribution. Spectral functions constructed in this way, therefore, are deter-mined by the probability distribution. They are probability spectral functions. Technolog-ically, in the scheme the state density of an eigenvalue spectrum is chosen as a probabilitydensity function, or equivalently, the spectral counting function of an eigenvalue spectrum ischosen as a cumulative probability function. Constructing the thermodynamic quality witha probability eigenvalue spectrum, we arrive at probability thermodynamics; constructingthe vacuum amplitude and the effective action, etc. with a probability eigenvalue spectrum,we arrive at a probability quantum field. – 1 –ur starting point is the heat kernel. Other spectral functions can be achieved fromthe heat kernel [2, 3, 5]. The local heat kernel K ( t ; r , r ′ ) of an operator D is the Greenfunction of the initial-value problem of the heat-type equation: ( ∂ t + D ) K ( t ; r , r ′ ) = 0 with K (0; r , r ′ ) = δ ( r − r ′ ) [2]. The global heat kernel is the trace of the local heat kernel K ( t ; r , r ′ ) : K ( t ) = ∞ X n =0 e − λ n t . (1.1)Here { λ n } is the eigenvalue spectrum of the operator D , determined by the eigenequation Dφ n = λ n φ n . In thermodynamics, the heat kernel is the partition function; in quantumfield theory, the heat kernel is the vacuum amplitude. From the heat kernel, we can obtainall thermodynamic quantities in thermodynamics and effective actions and vacuum energiesin quantum field theory.In probability spectral functions, we encounter unbounded spectra. This is becausethere are two types of probability distributions: one includes only nonnegative randomvariables, corresponding to lower bounded spectra, and the other includes negative randomvariables, corresponding to lower unbounded spectra. The definition of the heat kernel,Eq. (1.1), applies only to lower bounded spectra { λ , λ , λ , · · · } which has a finite lowesteigenvalue λ . For lower unbounded spectra, the lowest eigenvalue tends to negative infinityand the sum in Eq. (1.1) often diverges, so we need the generalized definition of the heatkernel.It should be emphasized that the eigenvalue spectrum of a real physical system mustbe lower bounded, or else the world would be unstable. While, the system considered hereis a fictional physical system. Their eigenvalue spectra are not from a physical operator,but is required to satisfy a probability distribution artificially. The eigenvalue spectrumcorresponding to a probability distribution with negative random variables leads to lowerunbounded eigenvalue spectra. Therefore, unsurprisingly, some thermodynamic qualitiesmay not be positive. That is, the operator corresponding to probability distributions issometimes not a physically real operator.Though the eigenvalue spectrum corresponding to probability distributions is some-times not physically real, the probability eigenvalue spectrum has good behavior. This isbecause the probability distribution has good behavior, such as integrability. Therefore, forexample, the vacuum energy though always diverges in quantum field theory, it convergesfor probability eigenvalue spectra.The scheme suggested in the present paper also generalizes some concepts in probabilitytheory. Take the moment generating function as an example. In probability theory, themoment generating function does not always exist. In this paper, the generalized heatkernel is used to serve as a generalized moment generating function. In many cases, the newdefinition is definable when the definition of the moment generating function in probabilitytheory fails.When constructing quantum fields, in some cases, we encounter divergences. We removethe divergence by a renormalization procedure.– 2 –s examples, we consider some probability distributions, including distributions with-out and with negative random variables. The corresponding thermodynamics and quantumfields are presented.In section 2, we discuss the generalization of spectral functions. In section 3, the proba-bility spectral function is introduced. In sections 4 and 5, probability thermodynamics andprobability quantum field theory are constructed. In section 6, various kinds of probabilitythermodynamics and probability quantum field theory corresponding to various probabilitydistributions are constructed. The conclusions and outlook are provided in section 7. Spectral functions are determined by a spectrum { λ n } . In physics, the spectrum { λ n } arealways an eigenvalue spectrum of an lower bounded Hermitian operator D . Nevertheless,the eigenvalue spectrum we are interested in is given by a probability distribution, whichmay be lower unbounded. For a lower bounded spectrum, there is a finite lowest eigenvalue λ min . For a lower unbounded spectrum, however, the lowest eigenvalue tends to negativeinfinity: λ min → −∞ . There are some discussions on the generalized spectral functions[6, 7]. For completeness, we give a complete discussion on this issue. The conventional definition of the global heat kernel for a lower bounded spectrum { λ , λ , λ , · · · } ,by Eq. (1.1), is K ( t ) = P ∞ n =0 e − λ n t . For lower unbounded spectra, {−∞ , · · · , λ − , λ , λ , · · · } ,a natural and naive generalization of the heat kernel, following the original intention of theconventional definition of the heat kernel, which is a sum of e − λ n t over all eigenstates, is K ( t ) = ∞ X n = −∞ e − λ n t . (2.1)This is, however, ill defined, because the existence of the divergent lowest eigenvalue λ −∞ = −∞ brings a divergent term e − λ −∞ t in the summation.In order to seek a generalized heat kernel for taking lower unbounded spectra intoaccount, we employ the Wick rotation.Wick rotation performs the replacement t = − iτ . (In physics, the Wick rotationusually takes t = iτ . The reason why we take t = − iτ is that we want to apply the resultto probability theory.) After the Wick rotation , the definition (2.1) becomes K W R ( τ ) = ∞ X n = −∞ e iλ n τ , (2.2)where K W R ( τ ) denotes the Wick-rotated heat kernel.After the Wick rotation, we may obtain a finite result K W R ( τ ) for lower unboundedspectra {−∞ , · · · , λ − , λ , λ , · · · } . Nevertheless, the Wick-rotated heat kernel K W R ( τ ) is,of course, not the heat kernel we want. To obtain a heat kernel rather than a Wick-rotated– 3 –eat kernel, after working out the sum in Eq. (2.2), we perform an inverse Wick rotation τ = it . After the inverse Wick rotation, the Wick-rotated heat kernel returns back to theheat kernel we want. The heat kernel obtained this way is finite.In short, the generalized heat kernel for a lower unbounded spectrum {−∞ , · · · , λ − , λ , λ , · · · } is obtained by the following procedure: K ( t ) = ∞ X n = −∞ e iλ n τ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ = it = K W R ( it ) . (2.3)Such a procedure is essentially an analytic continuation, which allows us to achieve a finiteresult of a divergent series.In this procedure, there are two steps to achieve the well-defined generalized heat kernelfor lower unbounded spectra, Eq. (2.3). The first step, instead of the divergent summation P ∞ n = −∞ e − λ n t in the naive generalization of heat kernel (2.1), we turn to calculate a con-vergent Wick-rotated summation P ∞ n = −∞ e iλ n τ . The second step, we perform an inverseWick rotation to the already worked out result of the Wick-rotated sum. Clearly, the keystep in this procedure is to convert the divergent sum P ∞ n = −∞ e − λ n τ to a convergent sum P ∞ n = −∞ e iλ n τ by the Wick rotation.Now we arrive at a definition (2.3) which defines a generalized heat kernel which isvalid for lower unbounded spectra. In the above, the generalized heat kernel is expressed in a summation form. By means ofthe state density ρ ( λ ) = X { λ n } δ ( λ − λ n ) , (2.4)we can rewrite the generalized heat kernel in an integral form. From the integral represen-tation, we can see how the Wick rotation treatment works more intuitively. Conventional heat kernel.
First, let us consider the conventional definition of heatkernels, Eq. (1.1), which is only valid for lower bounded spectra.By the state density (2.4), we can convert the sum in the conventional definition (1.1)into an integral, K ( t ) = Z ∞ dλρ ( λ ) e − λt . (2.5)The equivalence between the summation form (1.1) and the integral form (2.5) can bechecked directly by substituting the state density (2.4) into Eq. (2.5): K ( t ) = P { λ n } R ∞ dλδ ( λ − λ n ) e − λt = P ∞ n =0 e − λ n t .For lower bounded spectra, the integral representation (2.5) of heat kernels shows thatthe relation between the heat kernel and the state density is a Laplace transformation. Naive generalized heat kernel.
Next, let us see the naive generalization of heat kernels,Eq. (2.1). – 4 –onverting the sum in Eq. (2.1) into an integral gives K ( t ) = Z ∞−∞ dλρ ( λ ) e − λt . (2.6)Notice that for lower unbounded spectra, { λ n } runs from −∞ to + ∞ .For lower unbounded spectra, by inspection of Eq. (2.6), we can see that the rela-tion between the heat kernel and the state density is a two-sided Laplace transformation(also known as the bilateral Laplace transformation) [8] rather than a Laplace transforma-tion. The relation between a two-sided Laplace transformation and Laplace transformationis BL [ f ( λ ) ; t ] = L [ f ( λ ) ; t ] + L [ f ( − λ ) ; − t ] , where BL [ f ( λ ) ; t ] denotes the two-sidedLaplace transformation and L [ f ( λ ) ; t ] denotes the Laplace transformation.It is worth to point out here that the integral form of the naive generalized heat kernel(2.6) is also valid for a certain kind of lower unbounded spectra, though the sum form ofthe naive generalized heat kernel, Eq. (2.1), is invalid for all kinds of lower unboundedspectra. This is because in the summation form of the generalized heat kernel, Eq. (2.1),the term e − λ −∞ t diverges when λ −∞ = −∞ ; while, in the integral form, Eq. (2.6), thevalidity of the definition relies on the integrability of the integral, R ∞−∞ dλρ ( λ ) e − λt . If theintegral is integrable, then Eq. (2.6) can serve as a definition of heat kernels for lowerunbounded spectra. The integrability condition, clearly, is that the state density, ρ ( λ ) ,must attenuate rapidly enough. In other words, the definition (2.6) is not valid for all kindsof lower unbounded spectra; it is only valid for ρ ( λ ) decreasing faster than e λt . Wick-rotated heat kernel.
Now, let us see the Wick rotated heat kernel, Eq. (2.2).Converting Eq. (2.2) into an integral gives K W R ( τ ) = Z ∞−∞ ρ ( λ ) e iλτ dλ. (2.7)The Wick rotated heat kernel K W R ( τ ) is obviously a Fourier transformation of the statedensity ρ ( λ ) .Now we can explain why the Wick rotated treatment is needed for lower unboundedspectra. Without the Wick rotation, the integral form of the naive generalization of the heatkernel, Eq. (2.6), is a two-sided Laplace transformation, and the integrability condition isthat the state density ρ ( λ ) decreases faster than e λt . Nevertheless, the Wick rotated heatkernel, as shown in Eq. (2.7), is a Fourier transformation of the state density ρ ( λ ) . Theintegrability condition then becomes that ρ ( λ ) is absolutely integrable, which is easier tobe satisfied than that in the two-sided Laplace transformation. Generalized heat kernel.
The generalized heat kernel can be finally obtained by per-forming an inverse Wick rotation to the Wick rotated heat kernel K W R ( τ ) . From Eq.(2.7), we can see that the state density ρ ( λ ) is an inverse Fourier transformation of theWick rotated heat kernel K W R ( τ ) , i.e., ρ ( λ ) = 12 π Z ∞−∞ K W R ( τ ) e − iλτ dτ. (2.8)– 5 –ubstituting into Eq. (2.6) gives K ( t ) = Z ∞−∞ dτ K W R ( τ ) (cid:20) π Z ∞−∞ dλe − iλ ( τ + it ) (cid:21) = Z ∞−∞ dτ K W R ( τ ) δ ( τ − it ) . (2.9)Then we arrive at a relation between the heat kernel and the Wick rotated heat kernel: K ( t ) = K W R ( it ) . (2.10)This is just the result given by the integral form of the definition of the generalized heatkernel, Eq. (2.3).In a word, the key idea of introducing the generalized heat kernel for lower unboundedspectra is an analytic continuation treatment through a Wick rotation. The spectral counting function is the number of eigenstates whose eigenvalue is smallerthan a given number λ [3, 5], N ( λ ) = λ n ≤ λ X λ , (2.11)where λ denotes the minimum eigenvalue. The spectral counting function is an importantspectral function, which is the starting point of the famous problem formulated by Kac”Can one hear the shape of a drum?” [1] The spectral counting function N ( λ ) has a directlyrelation with the global heat kernel K ( t ) , or, the partition function Z ( β ) [3]. In probabilitytheory, the spectral counting function corresponds the cumulative probability function.For lower unbounded spectra, as that of heat kernels, the definition (2.11) needs to begeneralized as N ( λ ) = λ n ≤ λ X −∞ . (2.12)At the first sight, this seems to be a wrong definition: if the spectrum is not lower bounded,usually, there are infinite number of states below the state with eigenvalue λ n < λ .There are two ways to generalize the definition of the spectral counting function tolower unbounded spectra. Directly convert the sum in Eq. (2.12) into an integral is the most straightforward way togeneralize the spectral counting function to lower unbounded spectra: N ( λ ) = Z λ −∞ ρ ( λ ) dλ. (2.13)Similarly to heat kernels, the divergence encountered in the sum is avoided by the treatmentof converting the sum into an integral. In this sense, the counting function is well-definedso long as the state density ρ ( λ ) is integrable.– 6 – .2.2 Heat kernel approach Alternatively, in Refs. [3, 4], we present a relation between heat kernels and countingfunctions for lower bounded spectra: K ( t ) t = R ∞ N ( λ ) e − λt dλ , i.e., the counting function N ( λ ) is a Laplace transformation of K ( t ) /t . For lower unbounded spectra, the lowesteigenvalue is −∞ , so, as that of heat kernel, we can directly generalize the relation between K ( t ) and N ( λ ) as K ( t ) t = Z ∞−∞ N ( λ ) e − λt dλ. (2.14)Clearly, this generalized counting function for lower unbounded spectra is a two-sidedLaplace transformation (bilateral Laplace transformation) of K ( t ) /t rather than a Laplacetransformation.Performing a Wick rotation to Eq. (2.14), t = − iτ , gives K ( − iτ ) − iτ = Z ∞−∞ N ( λ ) e iλτ dλ = F [ N ( λ ) ; τ ] . (2.15)Then the two-sided Laplace transformation in Eq. (2.14) is converted to a Fourier transfor-mation. Therefore, the counting function N ( λ ) can be immediately obtained by an inverseFourier transformation: N ( λ ) = F − (cid:20) K ( − iτ ) − iτ ; λ (cid:21) = 12 πi Z ∞−∞ K ( τ ) τ e iλτ dτ. (2.16)Eqs. (2.14) and (2.16) are relations between generalized heat kernels and generalizedcounting functions.It should be noted that, at first sight, the counting function of a lower unbounded spec-trum will diverge since there are infinite eigenvalues below the given number λ . However, ifinstead the naive definition of counting function, N ( λ ) = P λ n <λ , by the definition (2.13),we may arrive at a finite result in the case of probability. The spectral zeta function is important in quantum field theory, which is the basics in thecalculation of the one-loop effective action and the vacuum energy, etc. [2, 5]. In proba-bility, the spectral zeta function indeed corresponds to the logarithmic moment generatingfunction.The conventional definition of the spectral zeta function, which is valid only for lowerbounded spectra, reads ζ ( s ) = ∞ X n =0 λ − sn . (2.17)– 7 – .3.1 State density approach To generalize the spectral zeta function to lower unbounded spectra, similarly as the gen-eralized heat kernel, we convert the sum into an integral with the state density ρ ( λ ) : ζ ( s ) = Z ∞−∞ dλρ ( λ ) λ − s . (2.18)This definition is valid so long as state density ρ ( λ ) λ − s is integrable. The relation between the conventional heat kernel and the conventional spectral zeta func-tion is a Mellin transformation [2], ζ ( s ) = 1Γ ( s ) Z ∞ t s − K ( t ) dt = 1Γ ( s ) M [ K ( t ) ; s ] , (2.19)where M [ K ( t ) ; s ] denotes the Mellin transformation. It can be checked that such a relationalso holds for generalized heat kernels and generalized spectral zeta functions. Substitutingthe generalized heat kernel (2.6) into Eq. (2.19) gives ζ ( s ) = 1Γ ( s ) Z ∞ dtt s − (cid:20)Z ∞−∞ dλρ ( λ ) e − λt (cid:21) = Z ∞−∞ dλρ ( λ ) λ − s . (2.20) First introduce a spectral characteristic function defined as the Fourier transformation ofthe state density, f ( k ) = Z ∞−∞ ρ ( λ ) e ikλ dλ. (2.21)Constructing a representation of λ − s , λ − s = 1 i s Γ ( s ) Z ∞ e ikλ k s − dk = 1 i s Γ ( s ) M h e ikλ ; s i , (2.22)and substituting into Eq. (2.18) give ζ ( s ) = Z ∞−∞ ρ ( λ ) (cid:20) i s Γ ( s ) Z ∞ e ikλ k s − dk (cid:21) dλ = 1 i s Γ ( s ) Z ∞ (cid:20)Z ∞−∞ ρ ( x ) e ikx dx (cid:21) k s − dk. (2.23)By the definition of the spectral characteristic function (2.21), we arrive at ζ ( s ) = 1 i s Γ ( s ) Z ∞ f ( k ) k s − dk (2.24) = 1 i s Γ ( s ) M [ f ( k ) ; s ] . (2.25)– 8 – .3.4 Mellin transformation approach The spectral zeta can also be expressed as a Mellin transformation, M [ f ( t ) ; s ] = Z ∞ f ( t ) t s − dt. (2.26)Rewriting the expression of the spectral zeta function (2.18) as ζ ( s ) = Z −∞ p ( x ) x − s dx + Z ∞ p ( x ) x − s dx, (2.27)we can representation the zeta by the Mellin transformation as ζ ( s ) = ( − − s − M [ − xp ( − x ) ; − s ] + M [ xp ( x ) ; − s ] . (2.28) The main aim of the present paper is to introduce probability spectral functions, whichbridges spectral theory and probability theory.The key idea to construct a probability spectral theory, including probability thermo-dynamics and probability quantum fields, is to regard random variables as eigenvalues.In probability theory, there are two kinds of probability distributions:(1) the random variable ranges from to ∞ ;(2) the random variable ranges from −∞ to ∞ .When regarding random variables as eigenvalue spectra, the distribution with random vari-ables ranging from to ∞ corresponds to the conventional heat kernel which is appropriatefor lower bounded spectra, and the distribution with random variables ranging from −∞ to ∞ corresponds to the generalized heat kernel which is appropriate for lower unboundedspectra.After bridging probability theory and the spectral theory by regarding random variablesas eigenvalues, we can further construct a fictional physical system whose eigenvalues obeya probability distribution. Thermodynamics and quantum fields of such a system can beestablished.Concretely, for a probability distribution, by regarding the eigenvalue λ as the randomvariables x , regarding the state density ρ ( λ ) as the probability distribution function p ( x ) ,and regarding the counting function as the cumulative probability functions P ( x ) , etc., wearrive at a set of probability spectral functions. In probability theory, there are variousprobability distributions, such as the Gaussian distribution, the Laplace distribution, stu-dent’s t -distribution, etc. For each probability distribution, we can construct a family ofspectral functions, including, e.g., the thermodynamic quantity and the effective action. In this section, we show that the similarity between spectral counting functions and thecumulative probability functions is a bridge between spectral theory and probability theory.– 9 –he spectral counting function is the number of eigenstates with eigenvalues less thanor equal to λ [3, 4]. In section 2.2, the definition of the spectral counting function isgeneralized to lower unbounded spectra. The generalized spectral counting function can beexpressed as N ( λ ) = Z λλ min ρ ( λ ) dλ, (3.1)where ρ ( λ ) is the state density and λ min tends to −∞ for lower unbounded spectra.In probability theory, there is a cumulative distribution function —— the probabilitythat a random variable will be found at a value less than or equal to λ [9]. The cumulativeprobability function can be expressed as P ( x ) = Z xx min p ( x ) dx, (3.2)where p ( x ) is the probability density function. The random variable x min tends to −∞ forprobability distributions with negative infinite random variables.Comparing the definition of the spectral counting function (3.1) and the cumulativeprobability functions (3.2), we can see that when regarding the random variable as aneigenvalue, the cumulative probability function serves as a spectral counting function.Along this line of thought, furthermore, we can also find the similarity between thestate density and the probability density function; the probability density function playsthe same role as the state density.The above observation builds a bridge between the probability theory and the spectraltheory. In the following, by regarding the random variable as an eigenvalue, we transformthe probability theory into a spectral theory. The moment generating function is an alternative description in addition to probabilitydensity functions and cumulative distribution functions. The moment generating function,however, unlike the characteristic function, does not always exist. In this section, we showthat the generalized heat kernel can serve as a generalized moment generating functionwhich is definable when the original definition of the moment generating function fails.In probability theory, the moment generating function M ( t ) is defined by a Riemann-Stieltjes integral [9], M ( t ) = Z ∞−∞ e tx dP ( x ) , (3.3)where P ( x ) is the cumulative probability functions.For continuous random variables the probability density function is p ( x ) . The momentgenerating function M ( − t ) is then a two-sided Laplace transformation of p ( x ) , M ( − t ) = Z ∞−∞ p ( x ) e − tx dx. (3.4)Observing the relation between the generalized heat kernel and the state density, Eq.(2.6), we can see that the generalized heat kernel K ( t ) = R ∞−∞ dλρ ( λ ) e − λt is just themoment generating function (3.4). – 10 –hen ρ ( λ ) decreases faster than e − λt , the moment generating function is ill-defined.Nevertheless, even for such a case, the generalized heat kernel is still well-defined after ananalytic continuation treatment.In a word, when replacing the moment generating function by the heat kernel, we infact introduce a generalized moment generating function instead of the original definition.The moment is fundamentally important in statistics: the first moment is the meanvalue, the second central moment is the variance, the third central moment is the skew-ness, and the fourth central moment is the kurtosis [10]. Moreover, the mean value of aquantity can be expressed as a series of the moments by expanding the quantity as a powerseries. Nevertheless, many statistical distributions have no moment, such as the Cauchydistribution and the intermediate distribution [11].By the generalized moment generating function, we can define moments for the statis-tical distributions which have no moments. In Ref. [11], for defining moments for statisticaldistributions, one introduces a weighted moment. When introducing the weighted moment,the moment depends not only on the distribution but also on the choice of weighted func-tion. The moment defined in the present paper is more natural, which depends only thestatistical distribution itself. In this section, we present the relation between the generalized heat kernel in spectraltheory and the characteristic function in probability theory.In probability theory, the characteristic function is defined as the Fourier transformationof the distribution function p ( x ) [10]: f ( k ) = F [ p ( x ) ; k ] = Z ∞−∞ dxp ( x ) e ikx , (3.5)where F [ p ( x ) ; k ] denotes the Fourier transformation of p ( x ) .The relation between the generalized heat kernel and the characteristic function canbe obtained by inspection of their definitions (2.6) and (3.5).As discussed above, by regarding the statistical distribution function p ( x ) as a statedensity ρ ( λ ) of a spectrum, we have K ( t ) = Z ∞−∞ dxp ( x ) e − xt . (3.6)From the definition of the characteristic function (3.5), we can see that the statistical dis-tribution function p ( x ) is the inverse Fourier transformation of the characteristic function: p ( x ) = 12 π Z ∞−∞ dkf ( k ) e − ikx . (3.7)Substituting Eq. (3.7) into Eq. (3.6) gives K ( t ) = Z ∞−∞ dkf ( k ) 12 π Z ∞−∞ dxe − i ( k − it ) x = Z ∞−∞ dkf ( k ) δ ( k − it ) . (3.8)– 11 –hen we arrive at a relation between the generalized heat kernel and the characteristicfunction: K ( t ) = f ( it ) . (3.9)It can be directly seen from Eq. (2.10) that the characteristic function is just the Wickrotated heat kernel introduced in section 2.1.2. Similarly, we can construct the probability zeta function. From the generalized spectral zetafunction (2.20), by regarding the eigenvalue λ as the random variables x and regarding thestatistical distribution function p ( x ) as the state density ρ ( λ ) , we arrive at a probabilityzeta function. ζ ( s ) = Z ∞−∞ dxp ( x ) x − s . (3.10)Based on this probability zeta function, we can obtain various spectral functions inquantum field theory, e.g., the effective action. All thermodynamic quantities are spectral functions. The thermodynamic behavior is de-termined by the spectrum. Starting from the partition function Z ( β ) , whether the conven-tional definition or the generalized definition, we arrive at thermodynamics. All thermody-namic quantities can be obtained from the partition function directly.Regarding the probability density function as a state density of an eigenvalue spectrum,each probability distribution, such as the Gaussian distribution and the Cauchy distribu-tion, defines a thermodynamic system. This allows us to establish various probabilitythermodynamics corresponding to various probability distributions. In statistical mechanics, the canonical partition function is defined by Z ( β ) = X { λ n } e − βλ n . (4.1)The canonical partition function is just the heat kernel with the replacement of t by β .By the same reason as in the definition of the heat kernel, the definition of the canonicalpartition function is only valid for lower bounded spectra. For lower unbounded spectra,as done for heat kernels, the canonical partition function can also be generalized.According to the generalization of the heat kernel given above, by Eqs. (2.10) and(2.7), we immediately achieve a generalized partition function, Z ( β ) = Z ∞ λ min ρ ( λ ) e − iλτ dλ (cid:12)(cid:12)(cid:12)(cid:12) τ = − iβ ; (4.2)or, by Eq. (2.6), we arrive at – 12 – ( β ) = Z ∞ λ min dλρ ( λ ) e − βλ . (4.3)For lower unbounded spectra, λ min → −∞ .It can be directly seen from Eq. (4.3) that, if regarding ρ ( λ ) e − βλ as a density function,the partition function is the zero-order moment (cid:10) λ (cid:11) .It should be emphasized that the partition function given by Eq. (4.2) or (4.3) is ageneralized partition function rather than the conventional partition function, and the cor-responding thermodynamic quantities are also not the conventional thermodynamic quan-tities. Starting from the generalized canonical partition function, Eq. (4.2) or (4.3), we canconstruct whole thermodynamics regardless of the spectrum lower bounded or lower un-bounded.The internal energy is a statistical average of eigenvalues, formally defined by U ( β ) = P { λ n } λ n e − λ n β P { λ n } e − λ n β . (4.4)For lower unbounded spectra, such a definition becomes U ( β ) = P ∞−∞ λ n e − λnβ P ∞−∞ e − λnβ . Thedivergence here is no longer a problem for it can be removed by the method discussedabove. It is worthy to point out that the internal energy for lower unbounded spectra maybe negative since part of the eigenvalues can take negative values.Observing the definition of the internal energy (4.4) and the partition function (4.1),we can formally write down the relation of internal energy and partition function even forlower unbounded spectra U ( β ) = − ∂∂β ln Z ( β ) = 1 Z ( β ) Z ∞ λ min dλρ ( λ ) λe − βλ . (4.5)From Eq. (4.5) we can see that the internal energy is the first-order moment (cid:10) λ (cid:11) .Furthermore, the specific heat can be calculated from the internal energy: C V = ∂U∂T = β R ∞ λ min dλρ ( λ ) λ e − λβ R ∞ λ min dλρ ( λ ) e − λβ − R ∞ λ min dλρ ( λ ) λe − λβ R ∞ λ min dλρ ( λ ) e − λβ ! . (4.6)It can be seen that the specific heat is the difference between the second-order moment andthe square of first-order moment (cid:10) λ (cid:11) − (cid:10) λ (cid:11) .Moreover, from the generalized canonical partition function, we can obtain other ther-modynamic quantities, e.g., the free energy F ( β ) = − β ln Z ( β ) (4.7)and the entropy S = ln Z ( β ) − β ∂∂β ln Z ( β ) (4.8)for both lower bounded and lower unbounded spectra.– 13 – .3 Probability thermodynamics: characteristic function approach Starting from the relation between the heat kernel and the characteristic function, wecan directly represent various thermodynamic quantities by the characteristic function.According to the correspondence between the generalized heat kernel and the canonicalpartition function (4.3) and the relation (3.9), we represent the canonical partition functionas Z ( β ) = f ( iβ ) . (4.9)Various thermodynamic quantities then can also be represented by the characteristic func-tion, e.g. the internal energy U ( β ) = − ∂∂β ln f ( iβ ) (4.10)and the specific heat capacity C V = β ∂ ∂β ln f ( iβ ) , (4.11)etc. The vacuum amplitude in the Euclidean quantum field theory is Z = Z D φe − I [ φ ] / ~ , (5.1)where I [ φ ] is the Euclidean action I [ φ ] = − R d xd ( it ) L . The Euclidean vacuum amplitudein spectral representation is just the partition function, or, the global heat kernel (1.1).The conventional definition of the vacuum amplitude (5.1) is, of course, only validfor lower bounded spectra. The procedure discussed above can also provide a generalizeddefinition of vacuum amplitude for lower unbounded spectra. This allows us to construct aquantum field for lower unbounded operators.By regarding the probability partition function as a probability vacuum amplitude, wearrive at a probability quantum field. Taking the one-loop effective action and the vacuumenergy as examples, we show how to construct a probability quantum field. The information of a mechanical system is embedded in an Hermitian operator D . Manyimportant physical quantities are spectral functions constructed from the eigenvalues { λ n } of the operator D . Two important quantities in quantum field theory, the one-loop effectiveaction and the vacuum energy, are both spectral functions: the one-loop effective action isthe determinant of the operator, det D , and the vacuum energy is the trace of the operator, tr D .For an Hermitian operator, the determinant is the product of the eigenvalues: det D = Y n λ n , (5.2)– 14 –nd the trace is the sum of the eigenvalues: tr D = X n λ n . (5.3)They are, obviously, divergent.In order to obtain a finite result, we need a renormalization treatment. The effective action for the operator D can be expanded as Γ [ φ ] = I [ φ ] + ~
12 ln det D + O (cid:0) ~ (cid:1) , (5.4)where W = 12 ln det D = 12 ln Y n λ n = 12 X n ln λ n (5.5)is the one-loop effective action which diverges.In order to obtain a finite one-loop effective action, we analytically continue the sumin the one-loop effective action (5.5) by virtue of the spectral zeta function. By Eq. (2.17)we can see that the derivative of the spectral zeta function is ζ ′ ( s ) = − P n λ − sn ln λ n , so ζ ′ (0) = P n ln λ n and then W = − ζ ′ (0) . (5.6)In practice, the one-loop effective action for continuous spectra can be rewritten as W = − Z ∞−∞ ρ ( λ ) ln λdλ, (5.7)by use of ζ ′ (0) = P n ln λ n = R ∞−∞ ρ ( λ ) ln λdλ .Now, we show that, in probability quantum field theory, the expression (5.6) for one-loop effective action is often convergent, though it diverges in many cases of quantum fieldtheory.In quantum field theory, in order to remove the divergence, one introduces a regularizedone-loop effective action [2]: W s = − e µ s Γ ( s ) ζ ( s ) . (5.8)where e µ is a constant of the dimension of mass introduced to keep proper dimension of theeffective action. To remove the divergence in the one-loop effective action, we expand W s around s = 0 : W s = − ζ (0) 1 s − ζ ′ (0) + (cid:18) γ E − ln e µ (cid:19) ζ (0) . (5.9)After the minimal subtraction which drops the divergent term, we arrive at a renormalizedone-loop effective action, W ren = − ζ ′ (0) + (cid:18) γ E − ln e µ (cid:19) ζ (0) . (5.10)– 15 –y the definition (2.17), ζ (0) = P n is divergent. It is just the divergent ζ (0) cancels thedivergence in ζ ′ (0) .In probability quantum field theory, the spectra satisfy probability distributions, so ζ (0) = P n R ∞−∞ ρ ( λ ) dλ = 1 is the total probability which is of course finite andnormalized. For finite ζ (0) , choosing the constant e µ = e γ E , we return to Eq. (5.6).Moreover, it should be emphasized that in probability quantum field theory, ζ ′ (0) stillmay diverge in some cases. For continuous spectra, ζ ′ (0) = P n ln λ n = R ∞−∞ ρ ( λ ) ln λdλ .In probability theory, ρ ( λ ) is the probability distribution function. Obviously, for proba-bility distribution possessing moments, ζ ′ (0) is finite. For probability distribution withoutmoments, we need to analyze the integrability case by case.The one-loop effective action can also be calculated directly from the spectral charac-teristic function introduced in section 2.3.3.Representing the state density ρ ( λ ) by the inverse Fourier transformation of the spec-tral characteristic function, by Eq. (3.7), ρ ( λ ) = 12 π Z ∞−∞ f ( k ) e ikλ dk (5.11)and substituting into the expression of the one-loop effective action, Eq. (5.7), we arrive at W = Z ∞−∞ (cid:20) π Z ∞−∞ f ( k ) e ikλ dk (cid:21) ln λdλ = 12 π Z ∞−∞ f ( k ) (cid:20)Z ∞−∞ e ikλ ln λdλ (cid:21) dk. (5.12)The integral in Eq. (5.12) diverges. This integral, however, is a Fourier transformation of ln λ . After analytic continuation, we achieve a finite result: Z ∞−∞ e ikλ ln λdλ = π (cid:20) − γ E δ ( k ) + iπδ ( k ) − sgn ( k ) − k (cid:21) . (5.13)Then we have W = (cid:18) − γ E + iπ (cid:19) f (0) − Z ∞−∞ f ( k ) sgn ( k ) − k dk. (5.14)Therefore, W = (cid:18) − γ E + iπ (cid:19) f (0) + Z −∞ f ( k ) k dk. (5.15)This expression is useful when only the characteristic function of a probability distri-bution is known.In some cases, the integral term in Eq. (5.15) still diverges. However, the divergence canbe removed by a renormalization procedure [12]. We will illustrate such a renormalizationprocedure by taking the Cauchy distribution as an example in section 6.2.5. The vacuum energy is essentially a sum of all eigenvalues, or, the trace of the operator D : E = X n λ n . (5.16)– 16 –hysical operators are not upper bounded, so the vacuum energy diverges. In section 2.3,we suggest an approach to obtain an finite result of the trace of the operator.The trace of the operator D , Eq. (5.3), is a Dirichlet series defined by ∞ X n =1 a n λ sn whichconverges absolutely on the right open half-plane such that Re s > . A Dirichlet series canbe analytically continued as a spectral zeta function.In our problem, the spectrum may be not lower bounded. That is, the trace of theoperator encountered sometimes is a generalized Dirichlet series. By the procedure devel-oped in section 2.3, we can analytically continue such a generalized Dirichlet series as ageneralized spectral zeta function.Concretely, the vacuum energy is just a spectral zeta function: E = tr D = X n λ sn (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) s = − = ζ ( s ) | s = − = ζ ( − . (5.17)The trace, after being analytically continued as a spectral zeta function, ζ D ( − , is alreadyfinite. That is, it is renormalized.Similarly, in quantum field theory, the vacuum energy always diverges. However, itrecovers ζ ( − after a renormalization treatment.The vacuum energy can also be calculated from the characteristic function.The vacuum energy can be expressed by the state density, E = Z ∞−∞ ρ ( λ ) λdλ. (5.18)The state density can be expressed by the inverse Fourier transformation of the character-istic function, so by Eq. (3.7), E = ( − i ) ddk (cid:20)Z ∞−∞ ρ ( λ ) e ikλ dλ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) k =0 = ( − i ) df ( k ) dk (cid:12)(cid:12)(cid:12)(cid:12) k =0 . (5.19) Following the approach introduced above, we can construct probability thermodynamicsand probability quantum fields for various statistical distributions.There are two kinds of probability distributions: lower bounded and lower unbounded.For distributions with nonnegative random variables, such as the gamma distribution,the exponential distribution, the beta distribution, the chi-square distribution, the Paretodistribution, the generalized inverse Gaussian distribution, the tempered stable distribution,and the log-normal distribution, we can use the conventional definition of the spectralfunction.For distributions with negative random variables, such as the normal distribution, theskew Gaussian distribution, Student’s t -distribution, the Laplace distribution, the Cauchy– 17 –istribution, the q -Gaussian distribution, the intermediate distribution, we must use thegeneralized definition of the spectral function. For distributions with nonnegative random variables, probability thermodynamics andprobability quantum fields can be constructed by the conventional definition of spectralfunctions.
The Gamma distribution is used to describe nonstationary earthquake activity, the dropsize in sprays [13], and maximum entropy approach to H -theory [14], etc.The gamma distribution [15] p ( x ) = 1Γ ( a ) b − a e − x/b x a − , (6.1)recovers many distributions when taking the shape parameter a and the inverse scaleparameter b as some special values. For example, the exponential distribution is recoveredwhen a = 1 and b = 1 /λ and the normal distribution is recovered when a → ∞ .The cumulative distribution function, by Eq. (3.2), reads P ( x ) = Q (cid:16) a, , xb (cid:17) , (6.2)where Q ( a, z , z ) = Γ ( a, z , z ) / Γ ( a ) is the generalized regularized incomplete gammafunction with Γ ( a, z , z ) the generalized incomplete gamma function [16]. Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = (1 − ibk ) − a . (6.3)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = (1 + bt ) − a . (6.4)The thermodynamics corresponding to the gamma distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = (1 + bβ ) − a . (6.5)Then the free energy F = − β ln(1 + bβ ) − a , (6.6)the internal energy U = ba bβ , (6.7)– 18 –he entropy S = baβ bβ − a ln(1 + bβ ) , (6.8)and the specific heat C V = b β a (1 + bβ ) . (6.9) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = 1Γ( a ) b − s Γ( a − s ) . (6.10)The one-loop effective action by Eq. (5.6) reads W = 12 (cid:16) ψ (0) ( a ) + ln b (cid:17) , (6.11)where ψ ( n ) ( z ) is the n -th derivative of the digamma function.The vacuum energy can be obtained by Eq. (5.17): E = ab. (6.12) The exponential distribution is used to describe the central spin problem [17], dislocationnetwork [18], and the Anderson-like localization transition in SU (3) gauge theory [19].The probability density function of the exponential distribution is [15, 20] p ( x ) = λe − λx , (6.13)where λ > is the rate parameter.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = 1 − e − λx . (6.14) Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = (1 − ibk ) − a . (6.15)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = λt + λ . (6.16)The thermodynamics corresponding to the exponential distribution can be then con-structed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = λβ + λ . (6.17)– 19 –hen the free energy F = − β ln λβ + λ , (6.18)the internal energy U = 1 β + λ , (6.19)the entropy S = ββ + λ + ln λβ + λ , (6.20)and the specific heat C V = β ( β + λ ) . (6.21) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = λ s Γ(1 − s ) . (6.22)The one-loop effective action by Eq. (5.6) reads W = − ln λ − γ E , (6.23)where γ E ≃ . is Euler’s constant.The vacuum energy can be obtained by Eq. (5.17): E = 1 λ . (6.24) The beta distribution fits the data in the measurement of monoenergetic muon neutrinocharged current interactions [21] and describes the collective behavior of active particleslocally aligning with neighbors [22].The beta distribution [23] p ( x ) = (1 − x ) b − x a − B ( a, b ) , (6.25)is defined on the interval [0 , , where a and b are two positive shape parameters [20].The cumulative distribution function, by Eq. (3.2), is P ( x ) = I x ( a, b ) , (6.26)where I z ( a, b ) = B ( z, a, b ) /B ( a, b ) is the regularized incomplete beta function with B ( z, a, b ) the incomplete beta function [16]. – 20 – hermodynamics In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = F ( a ; a + b ; ik ) , (6.27)where F ( a ; b ; z ) is the Kummer confluent hypergeometric function.The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = F ( a ; a + b ; − t ) . (6.28)The thermodynamics corresponding to the beta distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = F ( a ; a + b ; − β ) . (6.29)Then the free energy F = − β ln F ( a ; b + a ; − β ) , (6.30)the internal energy U = a F ( a + 1; b + a + 1; − β )( a + b ) F ( a ; b + a ; − β ) , (6.31)the entropy S = aβ F ( a + 1; b + a + 1; − β )( a + b ) F ( a ; b + a ; − β ) + ln F ( a ; b + a ; − β ) , (6.32)and the specific heat C V = aβ ( a + b ) (cid:20) ( a + 1) F ( a + 2; b + a + 2; − β ) F ( a ; b + a ; − β ) ( a + b + 1) − a F ( a + 1; b + a + 1; − β ) ( a + b ) F ( a ; b + a ; − β ) (cid:21) . (6.33) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = Γ( b )Γ( a − s ) B ( a, b )Γ( b − s + a ) . (6.34)The one-loop effective action by Eq. (5.6) reads W = 12 h ψ (0) ( a ) − ψ (0) ( b + a ) i , (6.35)where ψ ( n ) ( z ) is the n -th derivative of the digamma function.The vacuum energy can be obtained by Eq. (5.17): E = aa + b . (6.36)– 21 – .1.4 Chi-square distribution The chi-squared distribution ( χ -distribution) is the distribution of a sum of the squaresof standard normal random variables. The chi-square distribution appears in neutrinoexperiments [24], measurements of microwave background [25], and statistical inference[20, 26] . The probability density function of the chi-squared distribution with ν degrees of free-dom is [20] p ( x ) = 1Γ ( ν/
2) 2 − ν/ e − x / x ν − . (6.37)The cumulative distribution function, by Eq. (3.2), reads P ( x ) = Q ( ν/ , , x/ , (6.38)where Q ( a, z , z ) is the generalized regularized incomplete gamma function, defined as Γ ( a, z , z ) / Γ ( a ) with Γ ( a, z , z ) the generalized incomplete gamma function [27]. Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = F (cid:18) ν − k (cid:19) + √ ik Γ ( ν/
2) Γ (cid:18) ν (cid:19) F (cid:18) ν − k (cid:19) . (6.39)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = F (cid:18) ν t (cid:19) − t √ ν + 1) / ν/ F (cid:18) ν + 12 ; 32 ; t (cid:19) . (6.40)The thermodynamics corresponding to the chi-squared distribution can be then con-structed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = 1 √ π Γ (cid:18) ν + 12 (cid:19) U (cid:18) ν , , β (cid:19) , (6.41)where U ( a, b, z ) is the confluent hypergeometric function.Then the free energy F = − β ln (cid:18) √ π Γ (cid:18) ν + 12 (cid:19) U (cid:18) ν , , β (cid:19)(cid:19) , (6.42)the internal energy U = 2 ν − ν ) U ( ν/ , / , β / (cid:26) √ (cid:18) ν + 12 (cid:19) (cid:20) ( ν + 1) β F (cid:18) ν + 32 ; 52 ; β (cid:19) +3 F (cid:18) ν + 12 ; 32 ; β (cid:19)(cid:21) − β Γ (cid:16) ν (cid:17) F (cid:18) ν β (cid:19)(cid:27) , (6.43)– 22 –he entropy S = β
12Γ ( ν/
2) Γ( ν ) U ( ν/ , / , β / × (cid:26) √ ν ) (cid:26) √ π (cid:20) β ( ν + 1) F (cid:18) ν + 32 ; 52 ; β (cid:19) + 3 F (cid:18) ν + 12 ; 32 ; β (cid:19)(cid:21)(cid:27) − (cid:16) ν (cid:17) U (cid:18) ν + 12 , , β (cid:19) (cid:20) ln π − (cid:18) Γ (cid:18) ν + 12 (cid:19) U (cid:18) ν , , β (cid:19)(cid:19)(cid:21) − β ν +2 Γ (cid:16) ν (cid:17) Γ ( ν/ F (cid:18) ν β (cid:19)(cid:27) , (6.44)and the specific heat C V = − β (cid:2) Γ ( ν/ F ( ν/ , / , β / − √ β Γ (( ν + 1) / F (( ν + 1) /
2; 3 / β / (cid:3) × (cid:26) (cid:26) √ (cid:18) ν + 12 (cid:19) (cid:20) β ( ν + 1) F (cid:18) ν + 32 ; 52 ; β (cid:19) + 3 F (cid:18) ν + 12 ; 32 ; β (cid:19)(cid:21) − βν Γ (cid:16) ν (cid:17) F (cid:18) ν β (cid:19)(cid:27) +3 (cid:20) Γ (cid:16) ν (cid:17) F (cid:18) ν β (cid:19) − √ β Γ (cid:18) ν + 12 (cid:19) F (cid:18) ν + 12 ; 32 ; β (cid:19)(cid:21) × (cid:26) √ β ( ν + 1)Γ (cid:18) ν + 12 (cid:19) (cid:20) β ( ν + 3) F (cid:18) ν + 52 ; 72 ; β (cid:19) + 15 F (cid:18) ν + 32 ; 52 ; β (cid:19)(cid:21) − ν Γ (cid:16) ν (cid:17) (cid:20) β ( ν + 2) F (cid:18) ν β (cid:19) + 3 F (cid:18) ν β (cid:19)(cid:21)(cid:27)(cid:27) . (6.45) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = Γ (( ν − s ) / s/ Γ ( ν/ . (6.46)The one-loop effective action by Eq. (5.6) reads W = 12 (cid:16) ψ (0) (cid:16) ν (cid:17) + ln 2 (cid:17) , (6.47)where ψ ( n ) ( z ) is the n -th derivative of the digamma function.The vacuum energy can be obtained by Eq. (5.17): E = ν. (6.48) The Pareto distribution was introduced by Parado in social science [28]. The Pareto distri-bution appears in Calabi-Yau moduli spaces [29], the Anderson localization with disorderedmedium [30], and the phenomenon of cumulative inertia in intracellular transport [31].The probability density function of the Pareto distribution is [20] p ( x ) = b α x − − α α, (6.49)– 23 –here b is the minimum value parameter and α is the shape parameter.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = 1 − (cid:18) bx (cid:19) α . (6.50) Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = αE α +1 ( − ibk ) , (6.51)where E n ( z ) is the exponential integral function.The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = αE α +1 ( bt ) . (6.52)The thermodynamics corresponding to the Pareto distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = αE α +1 ( bβ ) . (6.53)Then the free energy F = − β ln ( αE α +1 ( bβ )) , (6.54)the internal energy U = bE α ( bβ ) E α +1 ( bβ ) , (6.55)the entropy S = bβE α ( bβ ) E α +1 ( bβ ) + ln ( αE α +1 ( bβ )) , (6.56)and the specific heat C V = b β (cid:2) E α − ( bβ ) E α +1 ( bβ ) − E α ( bβ ) (cid:3) E α +1 ( bβ ) . (6.57) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = αs + α b − s . (6.58)The one-loop effective action by Eq. (5.6) reads W = 12 (cid:18) α + ln b (cid:19) , (6.59)where ψ ( n ) ( z ) is the n -th derivative of the digamma function.The vacuum energy can be obtained by Eq. (5.17): E = αα − b. (6.60)– 24 – .1.6 Generalized inverse Gaussian distribution The generalized inverse Gaussian distribution is proposed to study the population frequen-cies of species [32], which is also called the Halphen type A distribution [33]. Sichel uses thedistribution to construct the mixture of the Poisson distribution [34, 35]. Barndorff-Nielsenpoints out that the mixture of the normal distribution and the generalized inverse Gaussiandistribution is just the generalized hyperbolic distribution [36, 37]. The generalized inverseGaussian distribution appears in for example fractional Brownian motion [38] and inverseIsing problem [39].The probability density function of the generalized inverse Gaussian distribution is [20] p ( x ) = µ − θ x θ − K θ ( λ/µ ) exp (cid:18) − λ x (cid:18) x µ + 1 (cid:19)(cid:19) , (6.61)where µ > is the mean, λ > is the shape parameter, and θ is a real number. K n ( z ) isthe modified Bessel function of the second kind [16]. Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = (cid:18) − ikµ λ (cid:19) − θ/ K θ (cid:18) ( λ/µ ) q − ikµ λ (cid:19) K θ ( λ/µ ) . (6.62)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = (cid:18) µ tλ + 1 (cid:19) − θ/ K θ (cid:16)p λ (2 tµ + λ ) /µ (cid:17) K θ ( λ/µ ) . (6.63)The thermodynamics corresponding to the generalized inverse Gaussian distributioncan be then constructed. The partition function, by replacing t with β in the heat kernel,reads Z ( β ) = 1 K θ ( λ/µ ) (cid:18) λ βµ + λ (cid:19) θ/ K θ ( σ ) . (6.64)Then the free energy F = − β ln (cid:18) λ βµ + λ (cid:19) θ/ K θ ( σ ) K θ ( λ/µ ) ! , (6.65)the internal energy U = µK θ ( σ ) s λ βµ + λ K θ +1 ( σ ) , (6.66)the entropy S = βλK θ +1 ( σ ) σK θ ( σ ) + ln (cid:18) λ βµ + λ (cid:19) θ/ K θ ( σ ) K θ ( λ/µ ) ! , (6.67)– 25 –nd the specific heat C V = β λ σ " σ K θ +1 ( σ ) K θ ( σ ) + λ (cid:0) βµ − λ (cid:1) µ K θ +1 ( σ ) K θ ( σ ) + σ K θ +2 ( σ ) K θ ( σ ) − σ K θ − ( σ ) K θ ( σ ) +2 σ K θ − ( σ ) K θ ( σ ) + σ K θ − ( σ ) K θ ( σ ) − σ K θ − ( σ ) K θ +1 ( σ ) K θ ( σ ) + 2 λ µ + 8 θ + 4 βλ (cid:21) , (6.68)where σ = p λ (2 βµ + λ ) /µ .Note that the generalized inverse Gaussian distribution recovers the inverse Gaussiandistribution when θ = − . Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = µ − s K s − θ ( λ/µ ) K θ ( λ/µ ) . (6.69)The one-loop effective action by Eq. (5.6) reads W = 12 (cid:18) ln µ + 1 K θ ( λ/µ ) ∂K − θ ( λ/µ ) ∂θ (cid:19) , (6.70)where K n ( z ) gives the modified Bessel function of the second kind.The vacuum energy can be obtained by Eq. (5.17): E = µ K θ +1 ( λ/µ ) K θ ( λ/µ ) . (6.71) The tempered stable distribution is introduced in Ref. [40]. In literature, the distributionis mentioned as the truncated Levy flight [41], and the KoBoL distribution [42]. Barndorff-Nielsen and Shephard generalize the tempered stable distribution to the modified stabledistribution [43]. The tempered stable distribution appears in for example continuous-timerandom walks [44].The probability density function of the tempered stable distribution is [45] p ( x ; κ, α, b ) = e αb e − b /κ x πα /κ ∞ X k =1 ( − k − sin ( kπκ ) Γ ( kκ + 1) k ! 2 kκ +1 (cid:16) xα /κ (cid:17) − kκ − , (6.72)where α > , b ≥ , and < κ < . Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = exp (cid:16) ab − a (cid:16) b /κ − ik (cid:17) κ (cid:17) . (6.73)For κ = 1 / the tempered stable distribution reduces to the inverse Gaussian distribution.For the limiting case → , we obtain the Gamma distribution.– 26 –he heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = exp (cid:16) ab − a (cid:16) b /κ + 2 t (cid:17) κ (cid:17) . (6.74)The thermodynamics corresponding to the tempered stable distribution can be thenconstructed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = exp (cid:16) ab − a (cid:16) b /κ + 2 β (cid:17) κ (cid:17) . (6.75)Then the free energy F = − aβ h b − (cid:16) b /κ + 2 β (cid:17) κ i , (6.76)the internal energy U = 2 a (cid:16) b /κ + 2 β (cid:17) − κ κ, (6.77)the entropy S = 2 aβ (cid:16) b /κ + 2 β (cid:17) − κ κ + a h b − (cid:16) b /κ + 2 β (cid:17) κ i , (6.78)and the specific heat C V = − aβ (cid:16) b /κ + 2 β (cid:17) − κ ( κ − κ. (6.79) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = − − s e ab sin κ ∞ X k =1 ( − k Γ( kκ + 1)Γ( − s − kκ )( k − a k b k + s/κ . (6.80)The one-loop effective action by Eq. (5.6) reads W = − ∞ X k =1 π ( − a ) k b k e ab Γ( k ) h ln b sin(csc( πκk )) − ln 2 sin κ csc( πκk ) − sin κ csc( πκk ) ψ (0) ( − kκ ) i . (6.81)The vacuum energy can be obtained by Eq. (5.19): E = 2 aκb ( κ − /κ . (6.82) The log-normal distribution is a continuous probability distribution whose logarithm isnormally distributed. The Log-normal distribution appears in for example the global oceanmodels [46], particle physics [47], and the problem of dark matter annihilation [48].The probability density function of the log-normal distribution is [20] p ( x ) = 1 √ πxσ exp (cid:18) − ( − µ + ln x ) σ (cid:19) . (6.83)The logarithm of a log-normal distribution X with parameters µ and is normally dis-tributed. ln X has a mean µ and standard deviation . Then we can write X as X = exp ( µ + σZ ) (6.84)– 27 –ith Z a standard normal variable.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = 12 erfc (cid:18) µ − ln x √ σ (cid:19) , (6.85)where erfc ( x ) is the complementary error function given by erfc ( z ) = 1 − erf( z ) with erf( z ) the integral of the Gaussian distribution: √ π R z e − t dt . Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = ∞ X n =0 ( ik ) n n ! exp (cid:18) nµ + n σ (cid:19) . (6.86)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = ∞ X n =0 ( ik ) n n ! exp (cid:18) nµ + n σ (cid:19) . (6.87)The thermodynamics corresponding to the log-normal distribution can be then con-structed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = ∞ X n =0 ( − β ) n n ! exp (cid:18) nµ + n σ (cid:19) . (6.88)Then the free energy F = − β ln ∞ X n =0 ( − β ) n n ! exp (cid:18) nµ + n σ (cid:19) , (6.89)the internal energy U = − P ∞ n =0 − n ( − β ) − n n ! exp (cid:16) nµ + n σ (cid:17)P ∞ n =0 ( − β ) n n ! exp (cid:16) nµ + n σ (cid:17) , (6.90)the entropy S = ln ∞ X n =0 ( − β ) n n ! exp (cid:18) nµ + n σ (cid:19) − P ∞ n =0 − n ( − β ) n n ! exp (cid:16) nµ + n σ (cid:17)P ∞ n =0 ( − β ) n n ! exp (cid:16) nµ + n σ (cid:17) , (6.91)and the specific heat C V = β hP ∞ n =0 ( − n β n n ! exp (cid:16) nµ + n σ (cid:17)i × ∞ X n =0 ( − β ) n − ( n − e n σ / µn ∞ X n =0 ( − β ) n n ! e n σ / µn − " ∞ X n =0 − ( − β ) n − ( n − e n σ / µn . (6.92)– 28 – uantum field The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = exp (cid:18) s (cid:0) sσ − µ (cid:1)(cid:19) . (6.93)The one-loop effective action by Eq. (5.6) reads W = µ . (6.94)The vacuum energy can be obtained by Eq. (5.17): E = exp (cid:18) µ + σ (cid:19) . (6.95) In the above, we construct probability thermodynamics and probability quantum fields forprobability distributions with nonnegative random variables. For constructing probabilityspectral functions with nonnagative random variables, we can use the conventional definitionof spectral functions. For the distribution with negative random variables, however, theconventional definition is invalid, and we need to use the generalized definition given in thepresent paper.
The normal distribution is also known as the bell curve or the Gaussian distribution. Nor-mal distribution is the most important distribution in statistics. The normal distributionappears in almost every areas.The probability density function of the normal distribution is [15, 20] p ( x ) = 1 √ πσ exp (cid:18) − ( x − µ ) σ (cid:19) , (6.96)where µ is the mean of the distribution and σ is the standard deviation.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = 12 erfc (cid:18) µ − x √ σ (cid:19) . (6.97) Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = exp (cid:18) ikµ − k σ (cid:19) . The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = exp (cid:18) t (cid:0) tσ − µ (cid:1)(cid:19) . (6.98)– 29 –he thermodynamics corresponding to the normal distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = exp (cid:18) β (cid:0) βσ − µ (cid:1)(cid:19) . (6.99)Then the free energy F = µ − βσ , (6.100)the internal energy U = µ − βσ , (6.101)the entropy S = − β σ , (6.102)and the specific heat C V = β σ . (6.103) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = ( − s s/ √ πσ s (cid:26) [1 + ( − s ] Γ (cid:18) − s (cid:19) F (cid:18) s − µ σ (cid:19) + √ µ [( − s − σ Γ (cid:16) − s (cid:17) F (cid:18) s + 12 ; 32 ; − µ σ (cid:19)) , (6.104)where Γ ( z ) is the Euler gamma function and F ( α ; b ; z ) is the confluent hypergeometricfunction. When s is an integer, ζ ( s ) = 12 s/ √ πσ s (cid:26) [1 + ( − s ] Γ (cid:18) − s (cid:19) F (cid:18) s − µ σ (cid:19) + √ µ ( − s [1 − ( − s ] σ Γ (cid:16) − s (cid:17) F (cid:18) s + 12 ; 32 ; − µ σ (cid:19)) . (6.105)When s is an even number ζ ( s ) = 12 s/ √ πσ s Γ (cid:18) − s (cid:19) F (cid:18) s − µ σ (cid:19) (6.106)and when s is an odd number ζ ( s ) = − (1 − s ) / µ √ πσ s +1 Γ (cid:16) − s (cid:17) F (cid:18) s + 12 ; 32 ; − µ σ (cid:19) . (6.107)The one-loop effective action by Eq. (5.6) reads W = 14 ∂ F (cid:0) a, / , − µ / σ (cid:1) ∂z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a =0 + iπ erfc (cid:18) µ √ σ (cid:19) + ln σ − γ E ! . (6.108)The vacuum energy can be obtained by Eq. (5.17): E = µ. (6.109)– 30 – .2.2 Skewed normal distribution The skewed normal distribution is a generalization of the normal distribution with a shapeparameter which accounts for skewness. Azzalini introduces the density function of theskewed normal distribution [49]. In physics, the skewed normal distribution appears ingravitational wave problems [50] and in black hole problems [51].The probability density function of the Skewed Gaussian distribution is [49] p ( x ) = 1 √ πσ e − ( x − µ ) / ( σ ) erfc (cid:18) − α ( x − µ ) √ σ (cid:19) , (6.110)where α is the shape parameter, µ is the location parameter, and σ is the scale parameter.The Gaussian distribution is recovered when α = 0 .The cumulative distribution function, by Eq. (3.2), reads P ( x ) = 12 erfc (cid:18) − x − µ √ σ (cid:19) − T (cid:18) x − µσ , α (cid:19) , (6.111)where erfi ( z ) is the imaginary error function given by erfi ( z ) = − i erf ( iz ) , and T [ x, α ] isOwen’s T function given by T [ x, α ] = π R α e − x ( t ) / t dt for real α [52]. Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = exp (cid:18) − k σ ikµ (cid:19) (cid:20) i erfi (cid:18) αkσ √ √ α + 1 (cid:19)(cid:21) . (6.112)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = e t ( σ t − µ ) erfc (cid:18) ασt √ √ α + 1 (cid:19) . (6.113)The thermodynamics corresponding to the skewed normal distribution can be thenconstructed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = e β ( βσ − µ ) erfc (cid:18) ασβ √ α + 2 (cid:19) . (6.114)Then the free energy F = − (cid:0) βσ − µ (cid:1) + 1 β ln erfc (cid:18) ασβ √ α + 2 (cid:19) , (6.115)the internal energy U = µ − βσ + 2 σα √ π √ α + 2 exp (cid:0) − α σ β / (cid:0) α + 2 (cid:1)(cid:1) erfc (cid:16) βσα/ √ α + 2 (cid:17) , (6.116)– 31 –he entropy S = 2 αβσ exp (cid:0) − α σ β / (cid:0) α + 2 (cid:1)(cid:1) √ π √ α + 2 erfc (cid:16) βσα/ √ α + 2 (cid:17) + ln (cid:18) erfc (cid:18) αβσ √ α + 2 (cid:19)(cid:19) − β σ , (6.117)and the specific heat C V = α β σ e − α β σ /α +1 h √ παβσe α β σ / α +2 erfc (cid:16) βσα/ √ α + 2 (cid:17) − √ α + 1 i π ( α + 1) / erfc (cid:16) βσα/ √ α + 2 (cid:17) + β σ . (6.118) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = 1 √ πσ Z ∞−∞ e − ( x − µ ) / ( σ ) erfc (cid:18) − α ( x − µ ) √ σ (cid:19) x − s dx. (6.119)The one-loop effective action by Eq. (5.7) reads W = 12 √ πσ Z ∞−∞ e − ( x − µ ) / ( σ ) erfc (cid:18) − α ( x − µ ) √ σ (cid:19) ln xdx. (6.120)The vacuum energy can be obtained by Eq. (5.19): E = µ + r π ασ √ α + 1 . (6.121)When µ = 0 , the integral in Eqs. (6.119) and (6.120) can be carried out, ζ ( s ) = 1 √ π σ − s (cid:26) s/ − (cid:2) ( − s − + 1 (cid:3) α s − Γ(1 − s ) ˜ F (cid:18) − s , − s − s − α (cid:19) + 2 − s/ ( − s Γ (cid:18) − s (cid:19)(cid:27) . (6.122)and W = 14 (cid:18) i cot − α + ln 2 σ + ψ (0) (cid:18) (cid:19)(cid:19) . (6.123)The vacuum energy can be obtained by Eq. (5.19): E = r π ασ √ α + 1 . (6.124) t -distribution Student’s t -distribution is a symmetric and bell shaped distribution, which rises to estimatethe mean of a normally distributed population whose sample size is small and populationstandard deviation is unknown. Student’s t -distribution has heavier tails which means thatit more likely to produce values that fall far from the mean. This is the main differencebetween the normal distribution and Student’s t -distribution.– 32 –he probability density function of Student’s t -distribution is [23] p ( x ) = 1 √ νσB (cid:0) ν , (cid:1) (cid:20) νν + ( x − µ ) /σ (cid:21) ( ν +1) / , (6.125)where µ is location parameter, σ scale parameter, and ν degrees of freedom. For ν → ∞ ,Student’s t -distribution goes to Gaussian distribution. For ν = 1 , Student’s t -distributiongoes to Cauchy distribution.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = I νσ x − µ )2+ νσ (cid:18) ν , (cid:19) , x ≤ µ, (cid:20) I ( x − µ )2( x − µ )2+ νσ (cid:18) , ν (cid:19) + 1 (cid:21) , x > µ, (6.126)where B ( a, b ) is the Euler beta function, K n ( z ) is the modified Bessel function of the secondkind, and I z ( a, b ) is regularized incomplete beta function given by I z ( a, b ) = B ( z, a, b ) /B ( a, b ) [16]. Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = 2 − ν/ ν ν/ σ ν/ Γ ( ν/ e ikµ | k | ν/ K ν/ (cid:0) √ νσ | k | (cid:1) . (6.127)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = 2 − ν/ ν ν/ Γ ( ν/ e − tµ ( σt ) ν/ K ν/ (cid:0) √ νσ | t | (cid:1) . (6.128)The thermodynamics corresponding to Student’s t -distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = 2 − ν/ ν ν/ Γ ( ν/ e − βµ ( βσ ) ν/ K ν/ (cid:0) β √ νσ (cid:1) . (6.129)Then the free energy F = − β ln 2 − ν/ ν ν/ e − βµ ( βσ ) ν/ K ν/ ( β √ νσ )Γ ( ν/ , (6.130)the internal energy U = 12 (cid:26) √ νσ K ν/ − ( β √ νσ ) + K ν/ ( β √ νσ ) K ν/ ( β √ νσ ) − νβ + 2 µ (cid:27) , (6.131)the entropy S = 12 " ln 4 ν ν/ e − βµ ( βσ ) ν K ν/ ( β √ νσ ) Γ ( ν/ + 2 βµ + 2 β √ νσK ν/ − ( β √ νσ ) K ν/ ( β √ νσ ) − ν ln 2 , (6.132)– 33 –nd the specific heat C V = − β νσ K ν/ ( β √ νσ ) (cid:8)(cid:2) K ν/ − (cid:0) β √ νσ (cid:1) + K ν/ (cid:0) β √ νσ (cid:1)(cid:3) − K ν/ (cid:0) β √ νσ (cid:1) (cid:2) K ν/ − (cid:0) β √ νσ (cid:1) + 2 K ν/ (cid:0) β √ νσ (cid:1) + K ν/ (cid:0) β √ νσ (cid:1)(cid:3)(cid:9) − ν . (6.133) Quantum field
The spectral zeta function, by Eq. (2.25), reads ζ ( s ) = √ π ( − ν/ e iπ ( ν +2 s ) / ν + s − (cid:18) iµ (cid:19) s Γ( s + ν )Γ ( ν/ ˜ F (cid:18) s , s + 12 ; 12 (2 s + ν + 1); νσ µ + 1 (cid:19) , (6.134)where ˜ F ( a, b ; c ; d ) is the regularized hypergeometric function and ˜ F ( a, b ; c ; d ) = F ( a, b ; c ; d ) / Γ ( c ) .The one-loop effective action by Eq. (5.6) reads W = ( − ν (cid:20) ln 4 µ − ψ (0) ( ν ) − iπ − Γ (cid:18) ν + 12 (cid:19) (cid:18) ˜ F (0 , , , (cid:18) , , ν + 12 , νσ µ + 1 (cid:19) × + ˜ F (0 , , , (cid:18) , , ν + 12 , νσ µ + 1 (cid:19) + ˜ F (1 , , , (cid:18) , , ν + 12 , νσ µ + 1 (cid:19)(cid:19)(cid:21) . (6.135)The vacuum energy can be obtained by Eq. (5.17): E = µ. (6.136) The Laplace distribution is known as the double exponential distribution that can be con-sidered as two exponential distributions jointed together back-to-back. The Laplace distri-bution is also a symmetric distribution. The Laplace distribution has fatter tails than thenormal distribution. The Laplace distribution appears in for example neuronal growth [53],complex networks [54], and complex biological systems [55].The probability density function of the Laplace distribution is [23, 56] p ( x ) = 12 e − λ | x − µ | λ, (6.137)where µ is the mean and λ is the scale parameter.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = − e − λ ( x − µ ) , x ≥ µ e λ ( x − µ ) , x < µ . (6.138) Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = λ k + λ e ikµ . (6.139)– 34 –he heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = λ λ − t e − tµ . (6.140)The thermodynamics corresponding to the Laplace distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = λ λ − β e − βµ . (6.141)Then the free energy F = − β ln λ λ − β e − βµ , (6.142)the internal energy U = 2 ββ − λ + µ, (6.143)the entropy S = (cid:18) βµ + 1 − β λ − β (cid:19) (cid:18) − βµ + ln λ λ − β (cid:19) , (6.144)and the specific heat C V = 2 β (cid:0) β + λ (cid:1) ( β − λ ) . (6.145) C V = 2 β (cid:0) β + λ (cid:1) ( β − λ ) . (6.146) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = 12 e − λµ µ − s (cid:16) − λµE s ( − λµ ) + e λµ ( λµ ) s Γ(1 − s, λµ ) − i sin( πs )Γ(1 − s )( λµ ) s (cid:17) , (6.147)where E n ( z ) gives the exponential integral functionThe one-loop effective action by Eq. (5.6) reads W = 14 e − λµ (cid:16) e λµ ln µ + Γ(0 , − λµ ) + e λµ Γ(0 , λµ ) + 2 iπ (cid:17) . (6.148)The vacuum energy can be obtained by Eq. (5.17): E = µ. (6.149) In statistics the Cauchy distribution is often used as an example of a pathological distribu-tion, for it does not have finite moments of order greater than one, but only has fractionalabsolute moments [20]. The Cauchy distribution appears in for example holographic dualproblems [57] and lasers [58].The probability density function of the Cauchy distribution is [20] p ( x ) = σπ [( x − µ ) + σ ] , (6.150)– 35 –here µ is location parameter and σ is scale parameter.The cumulative distribution function, by Eq. (3.2), reads P ( x ) = 1 π tan − (cid:18) x − µσ (cid:19) + 12 . (6.151) Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = e ikµ − σ | k | . (6.152)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = e − tµ − σ | t | . (6.153)The thermodynamics corresponding to the Cauchy distribution can be then constructed.The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = e − β ( µ + σ ) . (6.154)Then the free energy F = µ + σ (6.155)and the internal energy U = µ + σ. (6.156) Quantum field
The spectral zeta function, by Eq. (2.25), reads ζ ( s ) = ( µ + iσ ) − s . (6.157)The one-loop effective action by Eq. (5.6) reads W = 12 ln( µ + iσ ) . (6.158)The vacuum energy can be obtained by Eq. (5.17): E = µ + iσ. (6.159) Renormalization
In this section, we show that the renormalization procedure can re-move the divergence appears in the one-loop effective action and the result coincides withthe renormalized one-loop effective action obtain by the standard quantum field method.Consider a standard Cauchy distribution which is the distribution (6.150) with µ = 0 and σ = 1 , p ( x ) = 1 π ( x + 1) . (6.160)By Eq. (5.7), directly calculation gives W = Z ∞−∞ π ( λ + 1) ln λdλ = i π . (6.161)– 36 –he one-loop effective action can also be obtained by the characteristic function. Theresult can be obtained by Eq. (5.15) with a renormalization treatment.Substituting the distribution (6.160) into Eq. (5.15) gives W = (cid:18) − γ E + iπ (cid:19) f (0) + Z −∞ f ( k ) k dk. (6.162)The characteristic function of the standard Cauchy distribution is f ( k ) = e −| k | , (6.163)so W = − γ E + iπ Z −∞ e k k dk. (6.164)The integral diverges since k = 0 is singular. Cutting off the upper limit of the integralgives Z ε −∞ e k k dk = Ei ( ε ) , (6.165)where Ei ( z ) is the exponential integral function. Expanding at ε → , we arrive at Z ε −∞ e k k dk = Ei ( ε ) = γ E + ln ( − ε ) . (6.166)Dropping the divergent term ln ( − ε ) , we have Z −∞ e k k dk (cid:12)(cid:12)(cid:12)(cid:12) reg = γ E . (6.167)Then we obtain a renormalized one-loop effective action W | reg = − γ E + iπ Z −∞ e k k dk (cid:12)(cid:12)(cid:12)(cid:12) reg = iπ , (6.168)which agrees with Eq. (6.161).The validity of the above renormalization procedure is discussed in Ref. [12]. q -Gaussian distribution The q -Gaussian distribution is a generalization of the Gaussian distribution as well as thatthe Tsallis entropy is a generalization of the standard Boltzmann-Gibbs entropy or theShannon entropy [59]. The q -Gaussian distribution is an example of the Tsallis distributionarises from the maximization of the Tsallis entropy under some appropriate constraints.The probability density function of the q -Gaussian distribution is [60] p ( x ) = √ bC q e q (cid:0) − bx (cid:1) , (6.169)– 37 –here e q ( x ) = [(1 − q ) x + 1] − q is the q -exponential, q is deformation parameter, b ispositive real number, and the normalization factor C q is given by C q = √ π Γ (cid:16) − q (cid:17) (3 − q ) √ − q Γ (cid:16) − q − q ) (cid:17) , − ∞ < q < , √ π, q = 1 , √ π Γ (cid:16) − q q − (cid:17) √ q − (cid:16) q − (cid:17) , < q < , , (6.170)where x ∈ ( −∞ , + ∞ ) for ≤ q < and x ∈ (cid:18) − √ b (1 − q ) , + √ b (1 − q ) (cid:19) for q < with b thescale parameter and q the deformation parameter. For q = 1 , the q -Gaussian distributionreduces to the Gaussian distribution. For q = 2 and σ = q b , the q -Gaussian distributionreduces to the Cauchy distribution.The cumulative distribution function, taking < q < as an example, by Eq. (3.2),reads P ( x ) = x p b ( q − (cid:16) q − (cid:17) √ π Γ (cid:16) q − − (cid:17) F (cid:18) , q − − b ( q − x (cid:19) + 12 , < q < ,x √ b − bq Γ (cid:16) + − q (cid:17) √ π Γ (cid:16) − q (cid:17) F (cid:18) , q − − b ( q − x (cid:19) + 12 , q < , (cid:16) erf (cid:16) √ bx (cid:17) + 1 (cid:17) , q = 1 , , (6.171)where F ( a, b ; c ; z ) is the hypergeometric function and erf ( z ) is the error function. Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) : f ( k ) = 2 − q − q [ b ( q − q +14 − q + | k | − q − q − K q − q − | k | p b ( q − ! (cid:16) − q − q − (cid:17) . The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = 2 − q + (cid:2) b ( q − /t (cid:3) q − q − Γ (cid:16) q − − (cid:17) K + − q | t | p b ( q − ! . (6.172)The thermodynamics corresponding to the q -Gaussian distribution can be then con-structed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = 2 − q + (cid:20) b ( q − β (cid:21) q − q − K + − q β p b ( q − ! (cid:16) q − − (cid:17) . (6.173)– 38 –hen the free energy F = − β ln 2 − q + h b ( q − β i q − q − K + − q (cid:18) β √ b ( q − (cid:19) Γ (cid:16) q − − (cid:17) , (6.174)the internal energy U = K + − q (cid:18) β √ b ( q − (cid:19) + K q +12 − q (cid:18) β √ b ( q − (cid:19) p b ( q − K + − q (cid:18) β √ b ( q − (cid:19) + q − β ( q − , (6.175)the entropy S ( β ) = 14 √ b ( q − / K + − q (cid:18) β √ b ( q − (cid:19) (p b ( q − K + − q β p b ( q − ! × ( q −
1) ln − q + (cid:20) b ( q − β (cid:21) q − q − K + − q β p b ( q − ! / Γ (cid:18) q − − (cid:19)! + q − } + 2 β ( q − " K + − q β p b ( q − ! + K q +12 − q β p b ( q − ! + β ( q − p b ( q − K + − q β p b ( q − !) , (6.176)and the specific heat C V = q − q −
1) + β b ( q −
1) + β (cid:20) K − q − (cid:18) β √ b ( q − (cid:19) + K + − q (cid:18) β √ b ( q − (cid:19)(cid:21) b ( q − K + − q (cid:18) β √ b ( q − (cid:19) − β (cid:20) K + − q (cid:18) β √ b ( q − (cid:19) + K q +12 − q (cid:18) β √ b ( q − (cid:19)(cid:21) b ( q − K + − q (cid:18) β √ b ( q − (cid:19) . (6.177) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = [1 + ( − s ] Γ (cid:0) − s (cid:1) [ b ( q − s/ Γ (cid:16) s + q − − (cid:17) √ π Γ (cid:16) q − − (cid:17) . (6.178)When s is a integer ζ ( s ) = [1 + ( − s ] Γ (cid:0) − s (cid:1) [ b ( q − s/ Γ (cid:16) s + q − − (cid:17) √ π Γ (cid:16) q − − (cid:17) ; (6.179)– 39 –hen s is even ζ ( s ) = Γ (cid:0) − s (cid:1) [ b ( q − s/ Γ (cid:16) s + q − − (cid:17) √ π Γ (cid:16) q − − (cid:17) ; (6.180)when s is odd ζ ( s ) = [1 + ( − s ] Γ (cid:0) − s (cid:1) [ b ( q − s/ Γ (cid:16) s + q − − (cid:17) √ π Γ (cid:16) q − − (cid:17) = 0 . (6.181)The one-loop effective action by Eq. (5.6) reads W = 14 h − ln(4 b ( q − − H q − − + π i , (6.182)where H n gives the n th harmonic number.The vacuum energy can be obtained by Eq. (5.17): E = 0 . (6.183) The intermediate distribution is a distribution linking the Gaussian and the Cauchy dis-tribution [11], which reduces to the Gaussian and the Cauchy distribution at some specialvalues of the parameter. The intermediate distribution can be applied to spectral linebroadening in laser theory and the stock market return in quantitative finance [11].=The probability density function of the intermediate distribution is [11] p ( x, µ, σ, ν ) = 1 νπ e (1 /ν − σ Z cos (cid:18) x − µν ln t (cid:19) t σ ν − exp (cid:18)(cid:18) − ν (cid:19) σ t (cid:19) dt. (6.184) Thermodynamics
In order to obtain the thermodynamic quantity, we first calculatethe characteristic function by Eq. (3.5), which is a Fourier transformation of the densityfunction p ( x ) , f ( k ) = exp (cid:18) ikµ + (cid:16) − e − ν | k | (cid:17) (cid:18) − ν (cid:19) σ − σ | k | ν (cid:19) . (6.185)The heat kernel, by performing the Wick rotation k → it to the characteristic function,reads K ( t ) = exp − tµ + (cid:0) − e − ν | t | (cid:1) ( − ν ) σ ν − σ | t | ν ! . (6.186)The thermodynamics corresponding to the intermediate distribution can be then con-structed. The partition function, by replacing t with β in the heat kernel, reads Z ( β ) = exp (cid:18) − β (cid:18) µ + σ ν (cid:19) + σ ν ( ν − (cid:16) e − βν − (cid:17)(cid:19) . (6.187)Then the free energy F = µ + σ ν − ( ν − σ βν (cid:16) e − βν − (cid:17) , (6.188)– 40 –he internal energy U = µ + (cid:20) e − βν ( ν −
1) + 1 ν (cid:21) σ , (6.189)the entropy S = e − βν ( ν − βσ + ( ν − σ ν (cid:16) e − βν − (cid:17) , (6.190)and the specific heat C V = e − βν β ( ν − νσ . (6.191) Quantum field
The spectral zeta function, by Eq. (3.10), reads ζ ( s ) = 1 π ν − s sin( πs )Γ(1 − s ) exp − iπs σ (cid:0) − ν (cid:1) ν ! Z exp t σ (cid:0) ν − (cid:1) ν ! t iµν + σ ν − (ln t ) s − dt. (6.192) ζ ( s ) = 1 π ν − s sin( πs )Γ(1 − s ) exp − iπs σ (cid:0) − ν (cid:1) ν ! Z exp t σ (cid:0) ν − (cid:1) ν ! t iµν + σ ν − (ln t ) s − dt. (6.193)The one-loop effective action by Eq. (5.7) reads W = 12 e − σ (1 − /ν ) Z exp t σ (cid:0) ν − (cid:1) ν + (cid:18) iµν + σ ν − (cid:19) ln t ! t dt. (6.194)The vacuum energy can be obtained by Eq. (5.19): E = µ − iσ (cid:18) − ν − ν (cid:19) . (6.195) In this paper, we suggest a scheme for constructing probability thermodynamics and quan-tum fields. In the scheme, we suppose that there is a fictional system whose eigenvalues obeya probability distribution. Most characteristic quantities in thermodynamics and quantumfield theory are spectral functions, such as various thermodynamics qualities and effectiveactions.Our starting point is to analytically continue the heat kernel which is the partitionfunction in thermodynamics and the vacuum amplitude in quantum field theory. We notehere that the Wick-rotated heat kernel is a special case of the analytic continuation of theheat kernel, K ( z ) = ∞ X n = −∞ e λ n z , (7.1)which is defined in the complex plane with a complex number z . By Eq. (7.1), we analyti-cally continue the global heat kernel to the z -complex plane. When z is a pure imaginarynumber, the complex variable heat kernel, K ( z ) , returns to Eq. (2.2). By the complex– 41 –ariable heat kernel, K ( z ) , we can discuss the analyticity property of the heat kernel onthe complex z -plane.In the scheme, we suppose that the eigenvalue obeys a probability distribution withoutgiving the operator. In further work, we can consider what an operator corresponds aeigenvalue spectra obeying a certain given probability distribution and find operators forvarious probability distributions. As a rough example, the Gaussian distribution, p ( x ) = √ πσ e − ( x − µ ) / ( σ ) , corresponds to a Laplace operator ∆ . The negative Laplace operator − ∆ is lower bounded, corresponding to nonnegative random variables and the positiveLaplace operator +∆ is lower unbounded, corresponding to negative random variables.The work of this paper bridges two fundamental theories: probability theory and heatkernel theory. The heat kernel theory is important in both physics and mathematics. Inphysics, the heat kernel theory plays an important role in quantum field theory [2, 61–66].Scattering spectral theory is an important theory in quantum field theory [67–70]. Therelation between heat kernels and scattering phases bridges the heat kernel theory, thescattering spectral theory, and scattering theory [69, 71–73]. This inspires us to relate theprobability theory and the scattering theory which is an important issue in mathematicalphysics [74]. Acknowledgments
We are very indebted to Dr G. Zeitrauman for his encouragement. This work is supportedin part by NSF of China under Grant No. 11575125 and No. 11675119.
References [1] M. Kac,
Can one hear the shape of a drum? , The American Mathematical Monthly (1966), no. 4 1–23.[2] D. V. Vassilevich, Heat kernel expansion: user’s manual , Physics Reports (2003), no. 5279–360.[3] W.-S. Dai and M. Xie,
The number of eigenstates: counting function and heat kernel , Journal of High Energy Physics (2009), no. 02 033.[4] C.-C. Zhou and W.-S. Dai,
Calculating eigenvalues of many-body systems from partitionfunctions , Journal of Statistical Mechanics: Theory and Experiment (2018), no. 8083103.[5] W.-S. Dai and M. Xie,
An approach for the calculation of one-loop effective actions, vacuumenergies, and spectral counting functions , Journal of High Energy Physics (2010), no. 61–29.[6] D. Fursaev and D. Vassilevich,
Operators, Geometry and Quanta: Methods of SpectralGeometry in Quantum Field Theory . Theoretical and Mathematical Physics. SpringerNetherlands, 2011.[7] N. Birrell, N. Birrell, and P. Davies,
Quantum Fields in Curved Space . CambridgeMonographs on Mathematical Physics. Cambridge University Press, 1984. – 42 –
8] W. R. LePage,
Complex variables and the Laplace transform for engineers . CourierCorporation, 2012.[9] G. Grimmett and D. Welsh,
Probability: an introduction . Oxford University Press, 2014.[10] A. John,
Mathematical statistics and data analysis . Wadsworth & Brooks/Cole, 1988.[11] T. Liu, P. Zhang, W.-S. Dai, and M. Xie,
An intermediate distribution between Gaussian andCauchy distributions , Physica A: Statistical Mechanics and its Applications (2012),no. 22 5411–5421.[12] W.-D. Li and W.-S. Dai,
Scattering theory without large-distance asymptotics in arbitrarydimensions , Journal of Physics A: Mathematical and Theoretical (2016), no. 46 465202.[13] G. Michas and F. Vallianatos, Stochastic modeling of nonstationary earthquake time serieswith long-term clustering effects , Physical Review E (2018), no. 4 042107.[14] G. L. Vasconcelos, D. S. Salazar, and A. Macêdo, Maximum entropy approach to H-theory:Statistical mechanics of hierarchical systems , Physical Review E (2018), no. 2 022104.[15] S. M. Ross, Introduction to probability models . Academic press, 2014.[16] M. Abramowitz and I. A. Stegun,
Handbook of Mathematical Functions: With Formulars,Graphs, and Mathematical Tables , vol. 55. DoverPublications. com, 1964.[17] L. P. Lindoy and D. E. Manolopoulos,
Simple and accurate method for central spin problems , Physical review letters (2018), no. 22 220604.[18] R. B. Sills, N. Bertin, A. Aghaei, and W. Cai,
Dislocation networks and the microstructuralorigin of strain hardening , Physical review letters (2018), no. 8 085501.[19] T. G. Kovács and R. Á. Vig,
Localization transition in SU (3) gauge theory , Physical ReviewD (2018), no. 1 014502.[20] N. Johnson, S. Kotz, and N. Balakrishnan, Continuous univariate distributions, vol. 1 . JohnWiley & Sons Incorporated, 1995.[21] A. Aguilar-Arevalo, B. Brown, L. Bugel, G. Cheng, E. Church, J. Conrad, R. Cooper,R. Dharmapalan, Z. Djurcic, D. Finley, et al.,
First Measurement of Monoenergetic MuonNeutrino Charged Current Interactions , Physical review letters (2018), no. 14 141802.[22] E. Ó. Laighléis, M. R. Evans, and R. A. Blythe,
Minimal stochastic field equations forone-dimensional flocking , Physical Review E (2018), no. 6 062127.[23] N. L. Johnson, S. Kotz, and N. Balakrishnan, Continuous univariate distributions, vol. 2 ofwiley series in probability and mathematical statistics: applied probability and statistics .Wiley, New York„ 1995.[24] C. Das, J. Pulido, J. Maalampi, and S. Vihonen,
Determination of the θ
23 octant in longbaseline neutrino experiments within and beyond the standard model , Physical Review D (2018), no. 3 035023.[25] S. Aiola, B. Wang, A. Kosowsky, T. Kahniashvili, and H. Firouzjahi, Microwave backgroundcorrelations from dipole anisotropy modulation , Physical Review D (2015), no. 6 063008.[26] A. Mood, F. Graybill, and D. Boes, Introduction to the Theory of Statistics . McGraw-Hillinternational editions. Tata McGraw-Hill, 2001.[27] F. W. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark,
NIST handbook of mathematicalfunctions . Cambridge University Press, 2010. – 43 –
28] V. Pareto,
Cours d’économie politique , vol. 1. Librairie Droz, 1964.[29] C. Long, L. McAllister, and P. McGuirk,
Heavy tails in Calabi-Yau moduli spaces , Journal ofHigh Energy Physics (2014), no. 10 187.[30] A. A. Asatryan and A. Novikov,
Anderson localization of classical waves in weakly scatteringone-dimensional Levy lattices , Physical Review B (2018), no. 23 235144.[31] S. Fedotov, N. Korabel, T. A. Waigh, D. Han, and V. J. Allan, Memory effects and Lévy walkdynamics in intracellular transport of cargoes , Physical Review E (2018), no. 4 042136.[32] I. J. Good, The population frequencies of species and the estimation of populationparameters , Biometrika (1953), no. 3-4 237–264.[33] E. Halphen, Sur un nouveau type de courbe de fréquence , Comptes Rendus de l’Académie desSciences (1941) 633–635.[34] H. Sichel,
On a distribution representing sentence-length in written prose , Journal of theRoyal Statistical Society. Series A (General) (1974) 25–34.[35] H. S. Sichel,
On a distribution law for word frequencies , Journal of the American StatisticalAssociation (1975), no. 351a 542–547.[36] O. Barndorff-Nielsen, Exponentially decreasing distributions for the logarithm of particle size , Proc. R. Soc. Lond. A (1977), no. 1674 401–419.[37] O. Barndorff-Nielsen,
Hyperbolic distributions and distributions on hyperbolae , ScandinavianJournal of statistics (1978) 151–157.[38] A. Wyłomańska, A. Kumar, R. Połoczański, and P. Vellaisamy,
Inverse Gaussian and itsinverse process as the subordinators of fractional Brownian motion , Physical Review E (2016), no. 4 042128.[39] C. Donner and M. Opper, Inverse Ising problem in continuous time: A latent variableapproach , Physical Review E (2017), no. 6 062104.[40] M. Tweedie, An index which distinguishes between some important exponential families , in
Statistics: Applications and new directions: Proc. Indian statistical institute golden JubileeInternational conference , pp. 579–604, 1984.[41] I. Koponen,
Analytic approach to the problem of convergence of truncated Lévy flightstowards the Gaussian stochastic process , Physical Review E (1995), no. 1 1197.[42] S. I. Boyarchenko and S. Z. Levendorskiĺń, Option pricing for truncated Lévy processes , International Journal of Theoretical and Applied Finance (2000), no. 03 549–552.[43] O. E. Barndorff-Nielsen, N. Shephard, et al., Normal modified stable processes . MaPhySto,Department of Mathematical Sciences, University of Aarhus Aarhus, 2001.[44] T. Miyaguchi and T. Akimoto,
Ergodic properties of continuous-time random walks:Finite-size effects and ensemble dependences , Physical Review E (2013), no. 3 032130.[45] W. Schoutens, Lévy processes in finance, pricing derivatives , 2003.[46] B. Pearson and B. Fox-Kemper,
Log-Normal Turbulence Dissipation in Global Ocean Models , Physical review letters (2018), no. 9 094501.[47] B. Nachman and T. Rudelius,
A meta-analysis of the 8 TeV ATLAS and CMS SUSYsearches , Journal of High Energy Physics (2015), no. 2 4. – 44 –
48] M. Lisanti, S. Mishra-Sharma, N. L. Rodd, and B. R. Safdi,
Search for Dark MatterAnnihilation in Galaxy Groups , Physical review letters (2018), no. 10 101101.[49] A. Azzalini,
A class of distributions which includes the normal ones , Scandinavian journal ofstatistics (1985) 171–178.[50] J. R. Gair and C. J. Moore,
Quantifying and mitigating bias in inference on gravitationalwave source populations , Physical Review D (2015), no. 12 124062.[51] E. Torres-Lomas, J. C. Hidalgo, K. A. Malik, and L. A. Ureña-López, Formation ofsubhorizon black holes from preheating , Physical Review D (2014), no. 8 083008.[52] D. B. Owen, Tables for computing bivariate normal probabilities , The Annals ofMathematical Statistics (1956), no. 4 1075–1090.[53] D. J. Rizzo, J. D. White, E. Spedden, M. R. Wiens, D. L. Kaplan, T. J. Atherton, andC. Staii, Neuronal growth as diffusion in an effective potential , Physical Review E (2013),no. 4 042707.[54] P. S. Skardal, D. Taylor, and J. Sun, Optimal synchronization of complex networks , Physicalreview letters (2014), no. 14 144101.[55] S. Stylianidou, T. J. Lampo, A. J. Spakowitz, and P. A. Wiggins,
Strong disorder leads toscale invariance in complex biological systems , Physical Review E (2018), no. 6 062410.[56] S. Kotz, T. Kozubowski, and K. Podgorski, The Laplace Distribution and Generalizations: ARevisit with Applications to Communications, Economics, Engineering, and Finance .No. 183. Springer Science & Business Media, 2001.[57] T. Andrade, A. M. García-García, and B. Loureiro,
Coherence effects in disorderedgeometries with a field-theory dual , Journal of High Energy Physics (2018), no. 3 187.[58] E. Raposo and A. Gomes,
Analytical solution for the Lévy-like steady-state distribution ofintensities in random lasers , Physical Review A (2015), no. 4 043827.[59] C. Tsallis, Nonadditive entropy and nonextensive statistical mechanics-An overview after 20years , Brazilian Journal of Physics (2009), no. 2A 337–356.[60] S. Umarov, C. Tsallis, and S. Steinberg, On a q-central limit theorem consistent withnonextensive statistical mechanics , Milan journal of mathematics (2008), no. 1 307–328.[61] V. Mukhanov and S. Winitzki, Introduction to quantum effects in gravity . CambridgeUniversity Press, 2007.[62] A. Barvinsky and G. Vilkovisky,
Beyond the Schwinger-DeWitt technique: Converting loopsinto trees and in-in currents , Nuclear Physics B (1987) 163–188.[63] A. Barvinsky and G. Vilkovisky,
Covariant perturbation theory (II). Second order in thecurvature. General algorithms , Nuclear Physics B (1990), no. 2 471–511.[64] A. Barvinsky and G. Vilkovisky,
Covariant perturbation theory (III). Spectral representationsof the third-order form factors , Nuclear Physics B (1990), no. 2 512–524.[65] M. Beauregard, M. Bordag, and K. Kirsten,
Casimir energies in spherically symmetricbackground potentials revisited , Journal of Physics A: Mathematical and Theoretical (2015), no. 9 095401.[66] S. Rajeev, A dispersion relation for the density of states with application to the Casimireffect , Annals of Physics (2011), no. 6 1536–1547. – 45 –
67] N. Graham, M. Quandt, and H. Weigel,
Spectral methods in quantum field theory , vol. 777.Springer Science & Business Media, 2009.[68] S. J. Rahi, T. Emig, N. Graham, R. L. Jaffe, and M. Kardar,
Scattering theory approach toelectrodynamic Casimir forces , Physical Review D (2009), no. 8 085021.[69] H. Weigel, M. Quandt, and N. Graham, Spectral methods for coupled channels with a massgap , Physical Review D (2018), no. 3 036017.[70] N. Graham, Casimir energies of periodic dielectric gratings , Physical Review A (2014),no. 3 032507.[71] H. Pang, W.-S. Dai, and M. Xie, Relation between heat kernel method and scattering spectralmethod , The European Physical Journal C (2012), no. 5 1–13.[72] W.-D. Li and W.-S. Dai, Heat-kernel approach for scattering , The European Physical JournalC (2015), no. 6.[73] T. Liu, W.-D. Li, and W.-S. Dai, Scattering theory without large-distance asymptotics , Journal of High Energy Physics (2014), no. 6 1–12.[74] M. Reed and B. Simon,
Methods of modern mathematical physics, Vol. III: Scattering theory , New York, San Francisoco, London (1979).(1979).