Introduction to 5D optics for space-time sensors Christian J. Bordé Laboratoire de Physique des Lasers, Université Paris-Nord, Villetaneuse, France and SYRTE, Observatoire de Paris, Paris, France http://christian.j.borde.free.fr

Summary. — A new framework is proposed to compare and unify photon and atom optics, which rests on the quantization of proper time. A common wave equation written in …ve dimensions reduces both cases to 5D-optics of massless particles. The ordinary methods of optics (eikonal equation, Kirchho¤ integral, Lagrange invariant, Fermat principle, symplectic algebra and ABCD matrices....) are used to solve this equation in practical cases. The various phase shift cancellations, which occur in atom interferometers, and the quantum Langevin twin paradox for atoms, are then easily explained. A general phase-shift formula for interferometers is derived in …ve dimensions, which applies to clocks as well as to gravito-inertial sensors.

1. –Introduction Space-time sensors include atomic clocks and gravito-inertial sensors [1]. Most spacetime sensors today use either the interference of atom waves [2] or that of light beams. The same mathematical tools have been used to treat the propagation of both atoms and photons and to calculate phase shifts in interferometers: the WKB or eikonal approximation and the ABCD formalism [3]. One would like to go beyond the simple analogy and c Società Italiana di Fisica

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Christian J. Bordé

understand in depth what are the speci…c features brought by massive particles and what are the advantages of being able to modify the rest mass of the particles in interferometers [4, 5, 6]. The purpose of this course is to give a short introduction to generalizations of the WKB and ABCD formalisms for relativistic particles. A relativistic approach has the major advantage of providing a common framework for massive particles like atoms and massless ones like photons. A second step in this synthesis is to introduce mass as a quantum observable conjugate of proper time. Atom optics then becomes identical to photon optics in an extended (4+1)D space-time with 4 space dimensions. The optical path along the fourth dimension replaces the usual action phase factor in ordinary spacetime. Finally, within the approximation of a slowly varying phase and amplitude of the …eld, the dispersion surface can be locally approximated by a tangent paraboloid and the dynamics is that of a non-relativistic massive particle in all cases. The corresponding propagator gives the generalized ABCD law. This will lead us to a general formula for the phase shifts in interferometry. 2. –Klein-Gordon equation for matter waves

Atoms in a given internal energy state can be treated as quanta of a matter-wave …eld with a rest mass M corresponding to this internal energy and a spin corresponding to the total angular momentum in that state. To take this spin into account one can use, for example, a Dirac [7, 8, 9], Proca or higher-spin wave equation. Here, for simplicity, we shall ignore this spin and start simply with the Klein-Gordon equation for the covariant wave amplitude of a scalar …eld. The Lagrangian density for a complex scalar …eld ' is(1 ):

(1)

p L(x) = ~c g g

@ ' @ '

M 2 c2 + R ' ' ~2

The Euler-Lagrange equation (2)

@ [@L=@ (@ ')]

@L=@' = 0

yields the …eld equation: (3)

+

M 2 c2 + R '=0 ~2

(1 ) In this equation g is the determinant of the metric tensor g . The term R; where is an arbitrary numerical factor and R the Ricci scalar curvature, is a possible coupling between the …eld ' and gravitation. This term is of interest for future investigations in atom interferometry but in the remainder of these lecture notes we shall assume mimimal coupling and take = 0.

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Introduction to 5D optics for space-time sensors

where the d’Alembertian is related to the curved space-time metric g expression: (4)

'=g

r r ' = ( g)

1=2

@

h

1=2

( g)

g

@ '

by the usual

i

The canonical conjugate four-vector is: (5)

=

p @L = ~c gg @ (@ ')

@ '

and the conserved four-current density:

(6)

j =

i

p i~c gg

' + c:c: =

'@ ' + c:c:

The scalar product is thus:

(7)

('1 ; '2 ) =

i~

Z

g

p ! '1 @ '2 gd

An essential step in the formulation of the solution of the …eld equation is the derivation 0 of its Green function G(x; x ). Once this is achieved, boundary conditions and initial conditions can be introduced via a Kirchho¤-type representation of the solution (the magic rule of [10])

(8) '(x) =

Z

0

G(x; x )r0 '(x0 )

0

'(x0 )r0 G(x; x ) d

+

Z

0 p g (x0 )d4 x0 G(x; x )

assuming that the values of the …eld and its normal derivative are known on a spacelike hypersurface and where we have introduced a possible source term (x). The Green function and propagator of the Klein-Gordon equation are well-known in a ‡at space-time but have no simple expression with an arbitrary metric tensor. In what follows, we shall explore various approximations to solve this problem, such as the WKB or the quadratic approximations. 3. –Introduction of the proper time coordinate Previous equations assume that the massive particles have a unique well-de…ned mass M (9)

+

M 2 c2 '=0 ~2

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Christian J. Bordé

In the case of a composite complex body, such as an atomic species, there is a rich spectrum of internal energies and hence masses. This spectrum is the spectrum of eigenvalues of the internal Hamiltonian H0 . Thus we need to generalize the Klein-Gordon equation to include this variable mass aspect. For eigenstates of the mass operator this is achieved thanks to the ersatz: (10)

'(x; c ) = exp[i

M c2 ( ~

0 )]'(x; M c)

which obviously satis…es: (11)

i~

@'(x; c ) = c@

M c'(x; c )

where is a new quantum variable that we shall identify with the proper time. Rest mass and proper time thus appear as conjugate variables, which are both Lorentz invariants. So that, in the general case, the Klein-Gordon equation (9) can be written: 1 @2 '=0 c2 @ 2

b'

(12)

in which c is the only fundamental constant. Equation (12) is indeed the wave equation in a space with 4 spatial dimensions x; y; z; c [11]. We recover equation (9) in the case of a monomassic state for which the …eld oscillates at a single internal frequency M c2 =h(2 ). In the general case of a coherent superposition of mass states: (13)

1 '(x; c ) = p 2 ~

Z

d(M c) exp[i

Z

d (M c) < c jM c >< M cj' >

M c2 ( ~

0 )]'(x; M c)

or in Dirac notations: (14)

< c j' >=

The conjugate variables are respectively: the proper interval (15)

x4 =

x4 = c

p4 =

p4 = M c

and the mass (16)

(2 ) An interesting analogy is that of a laser pointer illuminating a perpendicular screen. The light beam in (3+1)D satis…es the wave equation with the velocity c: The spot on the screen satis…es a Klein-Gordon equation in (2+1)D where the mass term is given by the wave vector component imposed by the laser cavity resonator in the third space dimension. In the subspace of the screen the velocity is reduced in the range from 0 to c.

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Introduction to 5D optics for space-time sensors

dσ 2 = c 2 dt 2 − dx 2 − ds 2 = 0

t

s ds = c dt − dx 2 = 0 2

2

2

x

Fig. 1. –Elementary interval and generalized "light" cone in (4+1)D. Massive particles explore the additional space dimension s = c and have a velocity lower than c in space-time.

with the same relationship as between position and momentum: (17)

(pop )4 =

Mop c

E Mc

p

Fig. 2. – The dispersion surface is a cone in (4+1)D, which is projected in (3+1)D as a hyperboloid for each value of the mass.

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Christian J. Bordé

so that: (18)

(pop )^{ ' = i~@^{ ' with ^{ = 1; 2; 3; 4

(latin indices with a hat take the values 1,2,3,4 and greek ones the values 0,1,2,3,4). Proper time (interval) and mass thus become non-commuting observables and this is well illustrated by the gedanken photon box experiment of Einstein and Bohr, which plays the role of the Heisenberg microscope for these two quantities. In ‡at space-time, the propagator of equation (12) can be calculated, using standard techniques [10], either directly or from the Klein-Gordon propagator: (19)

~ ;T K (5) R;

1

1

=

2

(2 )

(c2 T 2

R2

c2

2 )3=2

Besides the mathematical di¢ culties in using this propagator, discussed in the 2D case in [10], it is not trivial to extend this result in the presence of arbitrary gravitational or inertial …elds. We shall therefore turn to approximate methods: the WKB propagator in the case of constant energy and the ABCD propagator for time-dependent systems. 4. –WKB solution We write the solution of the above …eld equations with a real amplitude and a real phase: (20)

a exp (i )

The coupled equations satis…ed by and a are respectively: - a generalized Hamilton-Jacobi equation: (21)

g

@

@

=

M 2 c2 a + ~2 a

g

@

and - a continuity equation:

(22)

h

@

1=2

( g)

i a2 = 0

which takes the familiar form: @ + div ( ~v ) = 0 @t

(23) with:

c = j0 =

1=2

2~c ( g)

g0 @

a2 and v i = j i =

1=2

2~c ( g)

gi @

a2 :

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Introduction to 5D optics for space-time sensors

If the quantum potential is neglected, the …rst equation becomes the usual HamiltonJacobi equation for massive particles in (3+1)D or as we shall see later an eikonal equation for massless particles in (4+1)D. The usual Hamilton-Jacobi equation is satis…ed by the classical action: (24)

S=

Z

Z

Ldt =

p dx

since (25)

@0 S =

p0

and

@i S =

pi

and (26)

g

p p = M 2 c2

From the relation between 4-momentum and 4-velocity (see the Appendix): x_

p = Mcp

(27)

g x_ x_

= M x_

one infers expressions for p20 and pi : p20 = g00 M 2 c2

(28) (29)

g00 M

2

_ i x_ j ij x

gij x_ j + gi0 x_ 0 = M

pi = M

_j ij x

+

gi0 p0 g00

where (30)

fij = gij

g0i g0j g00

is the 3D metric tensor. From p20 we get:

(31)

s

q

dt = M

i j ij dx dx =

p20 g00

M 2 c2 = M dl(3) =

s

p20 g00

M 2 c2

and the action is

(32) S =

Z

0

p0 dx

Z

i

pi dx =

p0 c(t

t0 ) +

Z

dl

(3)

s

p20 g00

M 2 c2

p0

Z

gi0 i dx g00

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Christian J. Bordé

so that the de Broglie wavelength is: (33)

dB

=q

h p20 g00

M 2 c2

In the case of (4+1)D the total phase

=p

h fij pi pj

= S (4) =~ satis…es the eikonal equation:

g ^ ^ @ ^ @^ = 0 with ^ ; ^ = 0; 1; 2; 3; 4

(34)

Equations (28), (31) become:

p20 =

(35)

(36)

dt =

M

p p0

g00 M 2 x_ 4 x_ 4

g00 q

fij dxi dxj

g00 M 2 fij x_ i x_ j

dx4 dx4 =

M

p p0

g00

dl(4)

and in the case of a constant energy the spatial part of the phase is now: (37)

Z

pi dxi + p4 dx4 = p0

Z p

fij dxi dxj dx4 dx4 p g00

p0

Z

gi0 i dx g00

The new optical path includes the path in the x4 part of space. The de Broglie wavelength becomes: (4) dB

(38)

=

h p0

and does not diverge any more for vanishing velocity. The above expression of the path provides a generalization of Fermat’s principle for the propagation of the waves associated with massive as well as massless particles:

(39)

Z

dl(4) p g00

gi0 i dx g00

=0

where the integral to be varied is taken between two points along the ray in 4 space dimensions. This Fermat’s principle proceeds from the existence of a Lagrange invariant I p^{ dx^{ in this enlarged space, consequence of the fact that p^{ is a gradient [15].

The photons propagate only in space-time where they have the maximum velocity c corrected by an index of refraction coming from g00 and also from fij (e.g. arising from the e¤ect of gravitational waves). Massive particules propagate also in the additional

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Introduction to 5D optics for space-time sensors

space dimension c and may thus have a reduced velocity in ordinary space (Figure 1). They accumulate phase shifts along the four space coordinates. The phase along c is not a¤ected by fij and by gi0 . Hence it will not be sensitive to gravitational waves or to rotation. This explains the reduced sensitivity of interferometers using non-relativistic particles to gravitational waves. Their enhanced sensitivity to rotation comes from the second term. The phase shift in atom interferometers may now be understood in terms of optical paths only, just as this is the case for ordinary optics in (3+1)D space-time. For example, the phase cancellation, which occurs between the contributions of the action and that of the separation of the end points in space [13, 1, 16, 17], is easily understood from the fact that these points lie on the same wave front in the extended 4D-space. Also the contribution to the recoil shift which originates from the action term in the phase [6, 14], has now an obvious interpretation as an optical path in the proper time dimension and constitutes a quantum realization of the Langevin twin paradox. The continuity equation can also be integrated as in the book of Born and Wolf [15]. We can write it as: a2

(40)

+@

@ a2 = 0

and if a2 is time independent:

(41)

@

! r

@ a2 = @ i @i a2 =

! ! Introducing the operator @ = rS r where along the beam, we obtain :

is a parameter which speci…es the position

@ ln a2 =

(42)

! 2 ra

S

From the Hamilton-Jacobi equation we get: (43)

@ S=

@ i S@i S = @ 0 S@0 S

2

M 2 c2 = g 00 (@0 S) + g 0i @0 S@i S

which gives dS = g 00 p20 + g 0i p0 pi

M 2 c2 d

When this is compared to 32:

(44)

dS =

s

p20 g00

M 2 c2 dl(3)

p0

gi0 i dx g00

M 2 c2

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Christian J. Bordé

one infers that d =q

(45)

dl(3) p20 g00

M 2 c2

From which we can integrate (42) along the ray: 3 2 Z (3) (3) S dl 5 a2 = a20 exp 4 q 2 p0 2 c2 M g00 for (3+1)D. Note that the d’Alembertian reduces to the Laplacian (with a minus sign) in the case of uniform rotation. When this calculation is repeated without mass but with hatted indices for (4+1)D one gets:

(46)

d^ =

p

g00 dl(4) dt = p0 M

and (47)

a2 = a20 exp

"Z

p

g00

b S (4) p0

dl(4)

#

This formula solves the problem of …nding the prefactor of the WKB propagator. 5. –Hamiltonian and Lagrangian expressions in the parabolic approximation . 5 1. Classical derivation. – First, we shall follow a simple track, starting with the approximations on classical formulas and turn later to quantum mechanics. From the classical relation between the 4-momentum p , the metric tensor g (with signature + ) and the rest mass of a particle M : (48)

g

p p = M 2 c2

and the relation between the covariant component p0 c (energy) and the contravariant one p0 c (relativistic mass times c2 ) : p0 c = g 00 p0 c + g 0i pi c

(49) we obtain (50)

p0

2

= g 00 (M 2 c2

pi f ij pj )

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Introduction to 5D optics for space-time sensors

where f ij = g ij

(51)

g 0i g 0j g 00

is the 3D metric tensor. The component p0 c is related to the relativistic mass M (see the Appendix) through p0 c = M c2

(52) and can be written p0 c =

(53)

M 2 c2 pi f ij pj g 00 M c2 + 2 2M

If M c2 is approximated by a known prescribed function of time, this formula remains valid to second-order (parabolic approximation) since: x=

x2 x0 + + O (x 2 2x0

x0 )2

and the Hamiltonian p0 c can be written as : (54) (55)

M c2 M 2 c2 + 00 2g 2M i; j = 1; 2; 3 H=

1 pi f ij pj 2M

g 0i pi c g 00

The Lagrangian is then:

(56)

L= =

pi x_ i H c2 M2 1 M g00 + M x_ i gij x_ j 2 M 2 1 g 0i g0i M cg0i x_ i M c2 00 2 g

In some cases it may be more convenient to assume that the energy E is close to a known value E0 either because energy is conserved and remains equal to its initial value or because of a slow variation of parameters. We can again make use of the identity: E2 E = E20 + 2E + O("2 ) valid to second-order in " = E E0 with either: 0 (57)

E 2 = p20 c2 =

c2 M 2 c2 g 00

pi g ij pj

2p0 g 0i pi

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Christian J. Bordé

or E 2 = p20 c2 = g00 M 2 c4

(58)

g00 c2 pi fij pj

In the parabolic approximation, the Hamiltonian can then be approximated by: E0 M 2 c4 + 2 2E0 g 00 E0 M 2 c4 ' + 2 2E0 g 00

(59)

H=

(60)

c2 pi g ij pj + 2p0 g 0i pi 2E0 g 00 c2 g 0i pi g ij pj + 00 pi c 00 2E0 g g

or by: (61)

H=

E0 M 2 c4 + g00 2 2E0

g00 c2

pi fij pj 2E0

This means that the usual hyperbolic dispersion curve is locally approximated by the parabola tangent to the hyperbola for the energy E0 [18]. This approximation scheme applies to massive as well as to massless particles (For example in the case of quasimonochromatic light M = 0 and p p E0 = ~! [3]). The non-relativistic limit is obtained for M ! M g 00 or E0 ! M c2 = g 00 . All these forms of the Hamiltonian can be shown to be equivalent thanks to (49). From the above Hamiltonians we can deduce a Schroedinger-like equation:

(62)

i~

@' M 2 c2 ~2 M c2 = + + @i f ij @j 00 @t 2g 2M 2M

i~c

g 0i @i ' g 00

which is identical to a Schroedinger equation for a non-relativistic particle of mass M (t) classical relativistic mass and with a shift in the rest mass. As we shall see in the next paragraph, this equation can also be derived directly from Klein-Gordon equation by a procedure analogous to that of H. Feshbach and F. Villars [12] in which the rest mass is replaced by the relativistic mass. . 5 2. Quantum mechanical derivation. –We generalize the procedure introduced by H. Feshbach and F. Villars [12] and a two-component wave function is de…ned through the combinations: = where (63)

1 =p 2

u v

'+ '

is a function of time only and where: =i

0

p = i~c gg 0 @ '

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Introduction to 5D optics for space-time sensors

The Klein-Gordon equation is equivalent to the set of two equations: g 0k 1 @k ' + p 00 g gg 00 p p = M 2 c4 g' + }2 c2 @i gf ij @j '

(64)

i}@t ' =

(65)

i}@t

i}c

i}c@k

g 0k g 00

where again g 0i g 0j g 00

f ij = g ij

(66) From which

i}@t

=

( +

(67)

3

+i 2

2)

M 2 c4

( 3 i 2) p 2 gg 00

i}c ( 2 i} + ( 2

p

i}c

0

1 ) @k

0

1 ) @t

g + }2 c2 @i

p

gf ij @j

g 0k @k g 00 g 0k g 00

where 0 ; 1 ; 2 ; 3 are the Pauli matrices. One could proceed with Foldy-Wouthuysen transformations, but we may also chose in order to decouple as much as possible large and small components. This requires that along the classical trajectory one should have, on the average: (68)

=

M

c2

1 p < g>

and for the large component we recover the Hamiltonian derived classically if the small corrections terms required for hermiticity and the term implying @t = are ignored. 6. –Schroedinger-like equation in (4+1)D

In (4+1) dimensions, the Hamiltonian becomes:

(69) (70)

M c2 1 p^{ f ^{|^p|^ 00 2g 2M ^{; |^ = 1; 2; 3; 4 H=

g 0^{ p^{ c g 00

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Christian J. Bordé

where we have introduced an extended metric tensor g ^ ^ (greek indices with a hat have integer values from 0 to 4: ^ ; ^ = 0; 1; 2; 3; 4 and latin ones from 1 to 4) such that g 44 = 1. Note that the components g 4 could be used to represent electromagnetic interactions as in Kaluza-Klein theory. In (4+1)D, with: (71)

(pop )^{ ' = i~@^{ '

the Schroedinger equation becomes:

(72)

i~

M c2 ~2 @' = + @^{ f ^{|^@|^ 00 @t 2g 2M

i~c

g 0^{ @^{ ' g 00

7. –Weak-…eld approximation In the weak-…eld approximation the space-time metric tensor takes the form (73) where

g

=

+h ;

jh j