Quantum mechanics, matter waves, and moving clocks

Quantum mechanics, matter waves, and moving clocks

Holger Müller
Abstract

This paper is divided into three parts. In the first (section 1), we demonstrate that all of quantum mechanics can be derived from the fundamental property that the propagation of a matter wave packet is described by the same gravitational and kinematic time dilation that applies to a clock. We will do so in several steps, first deriving the Schrödinger equation for a nonrelativistic particle without spin in a weak gravitational potential, and eventually the Dirac equation in curved space-time describing the propagation of a relativistic particle with spin in strong gravity.

In the second part (sections 2-4), we present interesting consequences of the above quantum mechanics: that it is possible to use wave packets as a reference for a clock, to test general relativity, and to realize a mass standard based on a proposed redefinition of the international system of units, wherein the Planck constant would be assigned a fixed value. The clock achieved an absolute accuracy of 4 parts per billion (ppb). The experiment yields the fine structure constant with 2.0 ppb accuracy. We present improvements that have reduced the leading systematic error about 8-fold and improved the statistical uncertainty to 0.33 ppb in 6 hours of integration time, referred to .

In the third part (sections 5-7), we present possible future experiments with atom interferometry: A gravitational Aharonov-Bohm experiment and its application as a measurement of Newton’s gravitational constant, antimatter interferometry, interferometry with charged particles, and interferometry in space.

We will give a review of previously published material when appropriate, but will focus on new aspects that haven’t been published before.

\@xsect

We will show that all of quantum mechanics can be derived from a picture of matter waves as clocks together with simple assumptions such as the principle of superposition. This picture assumes that a quantum mechanical wave packet has an oscillation frequency of , where is the particle’s mass, the velocity of light, and the reduced Planck constant. The oscillation frequency is shifted by the gravitational redshift and time dilation as the particle moves through space and time. The propagation of arbitary quantum states can be decomposed into such wave-packets (“matter-wave clocks”) taking all possible paths through phase-space. We will show that this path integral formalism will yield the quantum mechanical wave equations, starting with the Schrödinger equation for nonrelativistic, spinless particles, then for relativistic particles with spin, first without gravity, then in curved space-time. This shows that the picture of matter wave packets as Compton frequency clocks is not just exact. It can even be used to re-derive all of quantum mechanics.

The description of matter waves as matter-wave clocks has been the basis of de Broglie’s invention of matter waves [1]. It has recently been applied to tests of general relativity [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12], matter-wave experiments [13, 14, 15, 16, 17, 18, 19, 20, 21, 22], the foundations of quantum mechanics [23, 24], quantum space-time decoherence [25], the matter wave clock/mass standard [26, 29, 30], and led to a discussion on the role of the proper time in quantum mechanics [31, 32]. It is generally covariant and thus well-suited for use in curved space-time, e.g., gravitational waves [33, 34, 35, 36]. It has also given rise to a fair amount of controversy [37, 38, 39, 40, 41, 42, 43, 44, 45]. Within the broader context of quantum mechanics, however, this description has been abandoned, in part because it could not be used to derive a relativistic quantum theory, or explain spin.

The descriptions that replaced the clock picture achieve these goals, but do not motivate the concepts used. For example, the Dirac equation can be derived from a Lagrangian density, where takes the role of the coordinates: , where the are the Dirac matrices, the operator annihilates, and creates, a particle, and . This Lagrangian density is quadratic in and thereby allows to construct a path integral in Hilbert space. It, however, takes the existence of spinors and Dirac matrices for granted rather than explaining or motivating the need for them.

We shall construct a path integral directly from a Lagrangian that is a function of the space-time coordinates , where is the coordinate time, without making a nonrelativistic approximation or introducing additional fields. This will require us to introduce the Dirac matrices and spinors, and will thus explain their use. Since the phase accumulated by a wave packet is given by , it corresponds to a description of matter waves as clocks. We will thus arrive at a space-time path integral [46] in which is maintained exactly, that is equivalent to the Dirac equation.

This derivation shows that De Broglie’s matter wave theory naturally leads to particles with spin-1/2. It relates to Feynman’s search for a formula for the amplitude of a path in 3+1 space and time dimensions which is equivalent to the Dirac equation [47, 48]. It yields a new intuitive interpretation of the propagation of a Dirac particle and reproduces all results of standard quantum mechanics, including those supposedly at odds with it. Thus, it illuminates the role of the gravitational redshift and the proper time in quantum mechanics. Finally, we hope it offers an intuitive way to think about quantum mechanics and its possible generalizations.

\@xsect

We use letters from the second half of the Greek alphabet to denote the space-time coordinates. Letters from the second half of the Latin alphabet denote the spatial coordinates. In curved space-time, we shall employ both a coordinate frame with a metric and a local Lorentz frame with a Minkowski metric . The determinant of is denoted . Greek letters from the start of the alphabet will denote coordinates in the local Lorentz frame, the letters denote the spatial coordinates in the local Lorentz frame. The two frames are connected by the vierbein . Our Minkowski metric has a signature . The conventional Dirac matrices in the coordinate frame are and as well as and , where is the commutator. In weak gravitational fields, we write the metric as , where .

\@xsect

De Broglie started with Einstein’s equation and Planck’s , where is an energy, the mass of a particle, the velocity of light, the Planck constant, and a frequency [1]. The first relation implies that a massive particle has energy, and the second implies that a process having an energy is associated with an oscillation. The two relations together determine a frequency . That leads us to guess that maybe a particle is associated with an oscillation at that frequency. Since is related to the Compton wavelength by , we will call it the particle’s Compton frequency.

Naïvely, a particle moving at a velocity of could be described in two ways: The proper time measured by a co-moving clock for a moving reference frame is related to the coordinate time by , where . Consequently, the moving particle should accumulate fewer oscillations, as is replaced by . As measured by a clock at rest, we thus expect to observe a frequency

 ω′C=ωCdτdt=ωCγ−1. \hb@xt@.01(1)

However, one can make the converse argument: The energy of a moving particle is given by and should thus correspond to a frequency of

 ω′′C=ωCγ. \hb@xt@.01(2)

These seemingly contradictory results can be reconciled. For a wave, there are two velocities, phase velocity and group velocity . We assume the group velocity is identical to the classical velocity of the particle, . Thus, will determine the time dilation factor . The phase accumulated by the particle in its rest frame is . If a wave originates at then the same wave has the phase at a different location, where (by definition of ). We will try to determine such that this wave has the phase everywhere. In other words, we require

 ω′′Ct−k′′x=ω′Ct,k′′=ω′′Cvp. \hb@xt@.01(3)

We substitute and find

 ωCγ(1−vvp)=ωCγ, \hb@xt@.01(4)

which is solved by or . We have thus been able to overcome the first hurdle. A particle corresponds to an oscillation of frequency in its rest frame. Seen in the lab frame, it is a wave of frequency where is the total energy, group velocity , and phase velocity .

Let us denote the oscillation . Obviously, with hindsight we could identify it with the wave function, but we want to adopt a perspective that we do not know what it means just now. For example, we do not know whether it has to be a complex number, or how its amplitude is determined. We hope that these things will become clear when we know more about the wave’s behavior, and the theory will eventually be justified if it makes correct predictions for observable quantities. For now, we will speculate that, if the amplitude is high at a certain location, we will find a large number of particles there. We will adopt the latter point of view and defer the details for later study.) What we do know is that the phase of the wave is given by either the left or the right hand side of Eq. (Quantum mechanics, matter waves, and moving clocks), e.g.,

 ψ∝e−iωCτ. \hb@xt@.01(5)

A first experimentally observable effects can be deduced by studying the momentum of a particle. According to Eq. (Quantum mechanics, matter waves, and moving clocks),

 k=ω′′Cvp=mc2ℏγvc2=1ℏmγv \hb@xt@.01(6)

or

 p=ℏk. \hb@xt@.01(7)

This is de Broglie’s famous relation. It can be used to analyze, e.g., Young’s double slit experiment (using the principle of superposition).

\@xsect

So far, we can only analyze non-interacting particles, traveling on a straight line at constant velocity. We will gradually extend our formalism to study a particle in a potential and general trajectories. We assume we know and want to know , where and . Take a look at the double-slit experiment shown in Fig. 1, left). At some time between and , the particle has to pass through holes located at . Clearly, the contribution of to is given by the sum

 ψ(xB,tB)∝ψ(xA,tA)(e−iωCτ(A,1,B)+e−iωCτ(A,2,B)) \hb@xt@.01(8)

where is the proper time elapsed on the path from via to . The exact form of it is unimportant for now. If the screen has, say, holes located at , we obtain

 ψ(xB,tB)∝n∑n1=1ψ(xA,tA)e−iωCτ(A,n1,B). \hb@xt@.01(9)

What about many screens, each with many holes at , as shown in Fig. 1, right? Well,

 ψ(xB,tB)∝n∑n1=1n∑n2=1…n∑nN=1ψ(xA,tA)e−iωCτ(A,n1,n2,…,nN,B). \hb@xt@.01(10)

If each screen has an infinite number of holes and there are infinitely many screens, we obtain111With hindsight, by going from the sum without to the integral and thereby introducing the line elements , the interpretation of changed from a probability to a probability density.

 ψ(xB,tB)∝limN→∞∫dx1∫dx2…∫dxNψ(xA,tA)e−iωCτ(A,x1,x2,…,xN,B). \hb@xt@.01(11)

To evaluate the proper time , we split it up in sections . For each section,

 τ(xj,xj+1)=ϵ√1−v2j/c2, \hb@xt@.01(12)

where we used that is split into sections and is the velocity of the particle within that section, and . So,

 ψ(xB,tB)∝limN→∞∫dx1∫dx2…∫dxNψ(xA,tA)e−iϵωC⎛⎜⎝√1−v2Ac2+√1−v21c2+…√1−v2Nc2⎞⎟⎠. \hb@xt@.01(13)

In the exponent, we recognize the Riemannian sum and replace it by its limit, the integral

 −mc2ℏ∫dt√1−v2(t)c2=1ℏ∫Ldt=1ℏS,

where is the Lagrangian of a point particle in special relativity and the action. So we can write

 ψ(xB,tB)∝limN→∞∫dx1∫dx2…∫dxNψ(xA,tA)exp[iℏ∫dtL(x,˙x)] \hb@xt@.01(14)

or

 ψ(xB,tB)=∫Dxψ(xA,tA)exp[iℏ∫dtL(x,˙x)]. \hb@xt@.01(15)

The factor of in the Lagrangian is nothing but the relationship between proper time and coordinate time, . To include an interaction, we may use general relativity (GR), a description of gravity. The relationship between proper time and coordinate time in GR is

 dτ=√−gμνdxμdxν/c. \hb@xt@.01(16)

The Lagrangian of a point particle is still .

\@xsect

We shall follow the approach of Feynman [46]. We start by using the action

 S=−∫mc2√−gμνuμuνdt≈−∫mc2(1−12h00+h0jujc−12(δjk−hjk)ujcukc)dt \hb@xt@.01(17)

where we have expanded the square-root to leading order, choosing as a laboratory frame one in which the particle is moving slowly and the gravitational potential is weak.222The minus sign of comes from In this frame, is the usual 3-velocity. We now compute the path integral for an infinitesimal time interval and an infinitesimal distance . For an infinitesimal we have , so

 ψ(t+ϵ,(xA)j)=N∫d3qψ(t,(xA)j−qj)e−imc2ϵℏ(1−12h00)e−12Ajkqjqk+Bjqj \hb@xt@.01(18)

where is a normalization factor and

 Ajk≡−imℏϵ(δjk−hjk),Bj≡imcℏh0j. \hb@xt@.01(19)

We can expand in powers of :

 ψ+ϵ∂tψ \hb@xt@.01(20) =N∫d3q(ψ−qj∂jψ+12qjqk∂j∂kψ)(1−imc2ϵℏ(1−12h00))exp[12Ajkqjqk+Bjqj]

where . We compute

 ∫e−12Ajkqjqk+Bjqjd3q=(2π)3/2√detAe−12Bj(A−1)jkBk, \hb@xt@.01(21)

where is the determinant of and is the inverse matrix. We obtain

 ψ+ϵ∂tψ = N(2π)3/2√detA[(1−imc2ϵℏ(1−12h00))ψ \hb@xt@.01(22) −(∂jψ)∂∂Bj+12(∂j∂kψ)∂∂Bj∂∂Bk]exp(12BjBk(A−1)jk).

The normalization factor is determined from the fact that must approach for . We carry out the derivatives. We now neglect all terms that are suppressed by two powers of or more, including the terms, and terms proportional to . This leads to a Schrödinger equation

 iℏddtψ=−mc212h00ψ−ℏ22m(→∇−m→H)2ψ, \hb@xt@.01(23)

where we have substituted . The 3-vector is defined by .

To see that this is the familiar Schrödinger equation, we note that is the scalar gravitational potential. The significance of is a gravitational vector potential that describes “frame dragging” for a rotating source mass. This post-Newtonian effect of GR is extremely small on Earth.

From here on, we may derive the entire program of quantum mechanics, e.g., derive the conservation of the probability current to arrive at a interpretation of the wave function, the uncertainty relationship or commutation relations, and generalize the theory to describe multiple particles. This shows that quantum mechanics is a description of waves oscillating at the Compton frequency that explore all possible paths through curved spacetime.

\@xsect

The theory still has important gaps. We do not know about spin yet, and while we started relativistically, the Schrödinger equation we obtained is only nonrelativistic. It is not straightforward to obtain a relativistic theory in analogy to Eq. (15). The difficulties are substantial, so we will tackle them for a special relativistic framework, without gravity.

The difficulties arose when integrating the exponential over all of space, because there is no limit on the velocity . In particular, the integrand is not well behaved when and beyond. One might attempt to cut the integral before or anywhere else, but this would not lead to a Lorentz-invariant theory. The reason is that any speed below is the rest frame of a physically possible observer, and can thus not be excluded from the theory. Cutting at , on the other hand, doesn’t avoid divergence. Our luck in the previous chapter was that paths at and outside the light cone were suppressed by gaussian functions in the nonrelativistic framework. But now that we want to develop the relativistic theory, this is no longer possible. We are led to accept that the divergence is not a computational problem, but an indication that the model that we have used so far needs to be refined.

\@xsect

Since the difficulty arises from the square-root in the exponential, we shall try to avoid the square root. Using the momentum we shall re-write . The function , the Hamiltonian, turns out to be . We then use Dirac’s trick of replacing

 √p2c2+m2c4≡c(−→α)⋅→p+βmc2. \hb@xt@.01(24)

In order for this to work, we must require , and . (The sign of is arbitrary. We choose it to be negative, so that our end result has the familiar form.) It is clear that and cannot be ordinary numbers, but they may be matrices, e.g.,

 →α=(0→σ→σ0),β=(100−1), \hb@xt@.01(25)

where are the Pauli matrices. We now have

 L□=→p⋅˙→q+c→α⋅→p−mc2β. \hb@xt@.01(26)

Note that this Lagrangian is a matrix. For now, we shall continue our calculation and interpret this fact if and when we obtain a result.

We could now try inserting the new Lagrangian into the path integral, Eq. (15) and use . This, however, brings back the square-root and thus an integrand which is not well-behaved at the light cone. We can, however, generalize the path integral by treating as independent variables and integrate over all trajectories in phase-space, not just all trajectories in real space. We thus write

 ψ(→xB,tB)∫D3p1(2π)3∫D3xexp[−iℏ∫dt(→p⋅˙→q+c→α⋅→p−mc2β)]ψ(→xA,tA). \hb@xt@.01(27)
\@xsect

As before, consider an infinitesimal interval . We may use just one integration each. Noting that , we obtain

 ψ(t+ϵ,x)=N∫d3p(2π)3∫d3qexp[−iℏ→p⋅→q+iϵℏ(−c→α⋅→p+mc2β)]ψ(t,→x−→q). \hb@xt@.01(28)

We note that is given by the momentum-space wave function . Inserting this into the path integral gives

 ψ(t+ϵ,x)=−N∫d3p(2π)3exp[iϵℏ(−c→α⋅→p+mc2β)]e−i→p⋅→x/ℏΦ(−→p,t). \hb@xt@.01(29)

Since is an infinitesimal quantity, we may expand to first order on both sides of the equation:

 ψ(t,→x)+ϵ˙ψ(t,→x) \hb@xt@.01(30) =−N∫d3p(2π)3e−i→p⋅→x/ℏΦ(−→p,t)−N∫d3p(2π)3iϵℏ(−c→α⋅→p+mc2β)e−i→p⋅→x/ℏΦ(−→p,t).

The first term is the reverse Fourier transform and yields the position-space wave function. We determine the normalization factor by noting that if , the right hand side must equal the left hand side, i.e., . The remaining terms are

 ˙ψ(t,→x)=∫d3p(2π)3iℏ(−c→α⋅→p+mc2β)e−i→p⋅→x/ℏΦ(−→p). \hb@xt@.01(31)

We can replace the in the parenthesis by the derivative acting on the exponential,

 iℏ˙ψ=[ℏic→α⋅→∇+mc2β]ψ, \hb@xt@.01(32)

the Dirac equation!333I derived this on board the train to Varenna on July 14, 2013. We have thus arrived at a relativistic wave equation, and discovered spin. Our need to introduce the matrices and means the wave function is a vector having 4 components. We could now derive conserved quantities, find solutions to the Dirac equation, and recover the Schrödinger equation in the nonrelativistic limit. This would show us that the 4 components of are the particle and antiparticle with spin up and spin down, respectively.

Our notion of an elementary particle as a single clock turned out to be incompatible with relativity. Rather, a particle is a set of four clocks, two of which tick forward, two backward. The langrangian gives the time lags in an experiment comparing any of the four to another one.

\@xsect

We now come back to the interpretation: Let us label the spinor components of by an index . If a particle is found at four-position in a spin state , we may call this a spinor event . The components of the Lagrangian then represent the phase accumulated by the state between two infinitesimally separated spinor events and . The phase is, e.g., for , for , and if , where is an infinitesimal coordinate time interval. To calculate the phases between two events, the events have to be amended by a discrete coordinate .

The path integral Eq. (27) is over all of phase space, . Thus, there are arbitrary combinations of matrices in the exponential of one path, e.g., . Since each term with a matrix may change the spin , the particle not only takes all possible paths through phase space, but thereby also goes through all possible paths through spin space (Fig. 2). Loosely, we may draw an analogy between the propagation of a Dirac particle and observers carrying clocks on random paths through a building having four floors in which proper time passes at different rates - forward and backward. In such a building, time, geographical latitude and longitude as well as the floor level constitute a full description of an event .

We consider two special cases: (i) Eigenstates of ,

 eiL□/ℏdtψ=e−iωCdτψ=e−ipμdxμ/ℏψ, \hb@xt@.01(33)

are characterized by a definite momentum and do not change spin while propagating. The accumulated phase is equal to the proper time times the Compton frequency, i.e., the picture of matter waves as clocks applies exactly - not just in the nonrelativistic limit as before. (ii) A particle on a classical path extremizes its action. It will thus keep its spin state constant, as switching between such states (floor levels in the analogy) reduces the absolute value of the phase. Such particles can be treated without regard to spin and the phase accumulated along the path is .

\@xsect

To complete the demonstration that the clock picture and standard quantum mechanics follow from each other we outline how the clock picture can be derived from the Dirac equation. With , we see that

 |ψ(t+T)⟩=eiH(t1)ϵ/ℏ…eiH(tN)ϵ/ℏ|ψ(t)⟩, \hb@xt@.01(34)

where divide the interval in parts. Using position and momentum eigenstates with spin , we insert one each of the unity operators

 ∫dx∑s|→x,s⟩⟨→x,s|,∑s∫d3p(2π)3|→p,s⟩⟨→p,s| \hb@xt@.01(35)

between the exponentials. Noting that leads to Eq. (27).

\@xsect

The generalization to a particle in an electromagnetic field is straightforward by starting with the classical Lagrangian of a charged particle

 L=−mc2√1−v2/c2+ec→A→v−eΦ, \hb@xt@.01(36)

where the vector and scalar potential are differentiable but otherwise arbitrary functions of (there is no restriction to potentials that are at most quadratic in the coordinates as in nonrelativistic path integrals). Proceeding as above, we obtain

 →p = γm→v+ec→A,H=√(c→p−e→A)2+m2c4+eΦ, L□ = →p⋅→˙q−→α⋅(c→p−e→A)−mc2β−eΦ, \hb@xt@.01(37)

and again calculate a path integral over an infinitesimal interval , as in Eq. (27). This leads to the Dirac equation

 iℏ˙ψ=[c→α⋅(iℏ→∇−ec→A(→x))−mc2β−eΦ(→x)]ψ. \hb@xt@.01(38)

From the basic equations of motion, we could now proceed to construct the theory of interacting Fermions, i.e., quantum electrodynamics. Of course, this is a huge undertaking, requiring second quantization as a way of dealing with multi-particle systems. We will not consider this.

\@xsect\@xsect

The proper time is expressed by the Lagrangian

 L=dτdt=−mc√−gμν˙xμ˙xν. \hb@xt@.01(39)

The momentum is

 pμ=∂L∂˙xμ=mcgμν˙xν√−gμν˙xμ˙xν \hb@xt@.01(40)

and satisfies

 gκλpκpλ=m2c2gκλgκν˙xνgλμ˙xμ−gμν˙xμ˙xν=−m2c2. \hb@xt@.01(41)

We note that

 pμ˙xμ−L=mcgμν˙xν˙xμ√−gμν˙xμ˙xν−L=0. \hb@xt@.01(42)

Now we work in a specific frame and use

 H=pk˙xk−L=pμ˙xμ−p0˙x0−L=−p0˙x0=−p0c=−c√p20. \hb@xt@.01(43)

From Eq. (41), we obtain

 −m2c2=gμνpμpν=g00p0p0+2g0jp0pj+gjkpjpk, \hb@xt@.01(44)

which we may solve for and insert:

 H=c√1−g00(m2c2+2g0jp0pj+gjkpjpk) \hb@xt@.01(45)

At this point, let us define

 ¯gμν=gμν−g00,¯m2=m2−g00. \hb@xt@.01(46)

So that

 H=c√¯m2c2+2¯g0jp0pj+¯gjkpjpk. \hb@xt@.01(47)

In flat spacetime, this reduces to as it should. We note that under the square-root, so we have on the right hand side and the left hand side,

 H2=¯m2c4+2c¯g0jpjH+c2¯gjkpjpk. \hb@xt@.01(48)

We obtain

 H=c¯g0jpj±c√(¯g0j¯g0k+¯gjk)pjpk+¯m2c2. \hb@xt@.01(49)

We pick the plus sign so the Hamiltonian reduces to the usual one in flat space-time. We now introduce a dreibein so that

We define

 ¯αj=djaαa, \hb@xt@.01(51)

where are the familiar Dirac matrices. It is easy to check that

where denotes the anticommutator. Thus,

 (¯αjpj+β¯mc)2 = ¯αj¯αkpjpk+(¯αjβ+β¯αj)pj¯mc+β2¯m2c2 \hb@xt@.01(53) = 12(¯αj¯αk+¯αk¯αj)pjpk+β2¯m2c2 = (¯g0j¯g0k+¯gjk)pjpk+¯m2c2.

So we define

 L⋄=pk˙qk−c¯g0j(xk,t)pj−c[(−¯αj(xk,t))pj+β¯m(xk,t)c] \hb@xt@.01(54)

where we have explicitly denoted that the and depend on the coordinate and the time. (As before, the sign before is arbitrary and chosen such that the end result will reduce to the familiar Dirac equation in flat space time.) If all that works, our path integral will be

 ψ(t+T)=∫D3p(2π)3√−g∫D3x√−g×exp{∫[−iℏpkqk+icℏ¯g0jpj−icℏ(−¯αjpj+β¯mc)]dt}ψ(t,→x). \hb@xt@.01(55)

As before, we calculate an infinitesimal step

 ψ(→x,t+ϵ)=∫d3p(2π)3√−g∫d3q√−ge−iℏpkqkeicℏϵ¯g0jpje−icℏϵ[−¯αjpj+β¯mc]ψ(→x−→q,t). \hb@xt@.01(56)

Just as in the case without gravity, we are allowed to evaluate at instead of . That leaves us with

 ψ(→x,t+ϵ) = ∫d3p(2π)3eicℏϵ¯g0jpje−icℏϵ[−¯αjpj+β¯mc]∫d3qe−iℏpkqkψ(→x−→q,t) \hb@xt@.01(57) = ∫d3p(2π)3eicℏϵ¯g0jpje−icℏϵ[−¯αjpj+β¯mc]e−iℏpkxkΦ(−→p,t)

We use on the left hand side and obtain

 iℏ˙ψ = −∫d3p(2π)3c(¯g0jpj−[−¯αjpj+β¯mc])e−iℏpkxkΦ(−→p,t) \hb@xt@.01(58) = [ℏi(¯αj−¯g0j)∂j+β¯mc2]∫d3p(2π)3e−iℏpkxkΦ(−→p,t).

We are now able to write the Dirac equation in curved space-time in compact form

 iℏ˙ψ=