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PHYSICAL REVIEW D 86, 124017 (2012) Higher order of and

Claus La¨mmerzahl1,2,* and Patricia Rademaker1,3,† 1ZARM, University of Bremen, Am Fallturm, 28359 Bremen, Germany 2Institute for , University Oldenburg, 26111 Oldenburg, Germany 3Institute for Theoretical Physics, Leibniz University Hannover, Appelstrasse 2, 30167 Hannover, Germany (Received 1 September 2010; published 6 December 2012) Standard fundamental equations of motion for point are of second order in the . Here we are exploring the consequences of fundamental equations of motion with an additional small even higher order term to the standard formulation. This is related to two issues: (i) higher order equations of motion will have influence on the definition of the structure of possible interactions and in particular of the gravitational interaction, and (ii) such equations of motion provide a framework to test the validity of Newton’s second law which is the basis for the definition of but which assumes from the very beginning that the fundamental equations of motion are of second order. We will show that starting with our generalized equations of it is possible to introduce the -time metric describing the gravitational interaction by means of a standard gauge principle. Another main result within our model of even higher order is that for slowly varying and smooth fields the higher order derivatives either lead to runaway solutions or induces a zitterbewegung. We confront this higher order scheme with experimental data.

DOI: 10.1103/PhysRevD.86.124017 PACS numbers: 04.20.Cv, 04.20.Fy

I. INTRODUCTION the basic structure of the of motion. If, for ex- ample, the equation of motion is of higher than second The most basic approach to the mathematical descrip- order in the , then interactions could be tion of nature is provided by Newton axioms [1]. The most introduced in a different way and, thus, can have a different important of them state that (i) there are inertial systems, structure as we will show below. As a consequence, it is (ii) the of a body with respect to an inertial system is given by very important (i) to explore the structure of interactions and in particular of the gravitational interaction in the case mx€ ¼ Fðx; x_ ;tÞ; (1) of higher order fundamental equations of motion, and where m is the (inertial) of the body, x€ its accelera- (ii) to develop a theoretical framework for the description tion, and F the acting on the body which may depend of experiments testing the order of the equations of motion. on the and the of the , and that (iii) Equations of motion of higher order are related to a actio equals reactio [2]. different initial value problem: one needs more initial Leaving aside the fundamental and still unresolved prob- values beyond the initial position and initial velocity. In lem of how to really define an inertial system (see, e.g., physical terms this means that with respect to the time Ref. [3]), the second Newton axiom is a tool to explore the coordinate the equation of motion is more nonlocal than forces and, thus, to measure the fields acting on particles. The the ordinary second order equations. The extreme case that electromagnetic field, for example, can be explored and (formally) an infinite number of initial values are needed is measured through the observation of the acceleration of related to equations of motion with memory as, e.g., gen- charged particles under different conditions (different initial eralized Langevin equations where theR force equation pos- t ð 0Þ ð 0Þ 0 conditions, different charges, etc.). Thus, by means of the sesses an additional term of the form 0 C t t x_ t dt , of the form (1) the electromagnetic interaction and, see, e.g., Ref. [4]. in principle, all other interactions are defined. In many cases Also within quantum gravity scenarios it might be one uses quantum equations of motion like the Schro¨dinger expected that the effective equation of point particles or Dirac equation which, via the path integral approach or the may contain small higher order time derivatives. In fact, Ehrenfest theorem, for example, are also based on (1). All since space-time fluctuations in the sense of, e.g., fluctua- these considerations extend to relativistic equations of mo- tions of the metrical , yield Langevin-like equations tion. And all this is also the basis of the introduction of of motion also quantum gravity scenarios naturally are interactions through a gauge formalism which is basic for expected to lead to effective higher order equations of the theoretical description of the standard model. motion where the additional higher derivatives probably From this observation it is clear that what one defines as scale with, e.g., the Planck length. interaction or as the corresponding force field depends on Higher order derivatives in equations of motion occur in effective equations which take backreaction into account. *[email protected] One example for that is the Abraham-Lorentz equation for †[email protected] charged particles taking into account the electromagnetic

1550-7998=2012=86(12)=124017(10) 124017-1 Ó 2012 American Physical Society CLAUS LA¨ MMERZAHL AND PATRICIA RADEMAKER PHYSICAL REVIEW D 86, 124017 (2012) radiated away [5] or the radiation equation d 0 ¼ rL r L: (3) in gravity where the emitted gravitational waves are taken dt x_ into account, [6]. In the electromagnetic case the leading term is a third time derivative which leads to unphysical We obtain the same equation of motion from two runaway solutions which still is an unresolved problem. Lagrange functions if they differ by a total time derivative However, these equations discussed in relation with radia- only, tion damping are no fundamental equations, they are ef- d Lðt;x;x_ Þ!L0ðt;x;x_ Þ¼L ðt;x;x_ Þþ fðt;xÞ; (4) fective equations emerging from the fact that one no longer 0 dt regards the particles as test particles. Here we are only interested in the fundamental equations of motion. where the f is allowed to depend on t and x only. Since spin is some element of nonlocality it is not aston- We can expand the total time derivative, ishing that also the dynamics of spinning particles effectively 0ð x xÞ¼ ð x xÞþ ð xÞþxr ð xÞ can be described by means of a higher order theory [7]. L t; ; _ L0 t; ; _ @tf t; _ f t; : (5) Pais and Uhlenbeck [8] studied higher order mechanical models as a toy model for discussing properties of higher We are now invoking the gauge principle which pre- order field theories where higher order derivatives came in scribes the replacement, naturally in noncommutative models [9] or are introduced ð xÞ! ð xÞ r ! Að xÞ in order to eliminate divergences; see, e.g., Ref. [10]. @tf t; q t; ; f q t; ; (6) Recently, it has been shown in Ref. [11] that the of the Pais-Uhlenbeck oscillator is bounded from below, is where q is the coupling parameter (charge). The new ð xÞ Að xÞ unitary and is free of ghosts; see also Ref. [12] for further functions t; and t; are the and vector studies in this direction. potential of the Maxwell theory. Then the Lagrangian In the following we will consider fundamental equations coupled to these potentials reads of motion of even higher order which can be derived from a 0ð x xÞ¼ ð x xÞ ð xÞþ xAð xÞ . In order to be able to confront these L t; ; _ L0 t; ; _ q t; q _ t; : (7) modified equations of motion with experimental data, one ð x xÞ¼1 x2 first has to investigate the structure of interactions. This With the choice L0 t; ; _ 2 m _ we obtain the standard will be done by using a gauge principle for a second order equation of a charged particle moving in an Lagrange formalism (which in principle can be applied electromagnetic field. to Lagrangians of all orders). The solutions of the equa- Now we generalize this approach to higher order tions of motion coupled to these generalized gauge fields Lagrange functions. give—according to the chosen sign of the additional even higher derivative term—only runaway solutions or a B. Second order formalism zitterbewegung showing that in the latter case the standard In the second order formalism we consider Lagrange equations of motion are rather robust against addition of functions of the form higher order terms. We also discuss the experimental pos- ¼ ð x x xÞ sibilities to search for effects related to such higher order L L0 t; ; _ ; € (8) time derivatives. As an interesting by-product of the cor- responding higher order gauge formalism we obtain the from which we obtain the fourth order equation of standard space-time metric as an ordinary gauge field. motion, d d2 0 ¼ rL r L þ r L II. LAGRANGE FORMALISM x_ 2 x€ dt dt d d First we will use the Lagrange formalism in order to ¼ rL r L r L ; (9) introduce interactions into a theory containing higher order dt x_ dt x€ derivatives. We introduce interactions by means of a gauge r principle. We set up our notation by repeating shortly the where a denotes the gradient with respect to the variable standard first order formalism and then apply the gauge a. Also in this case we obtain the same equation of motion principle to a second order Lagrange formalism. from another Lagrange function if it differs from the original one by a total time derivative of a function f only. This function, however, now may depend on the A. First order formalism x_ : A Lagrange function of first order, ð x x xÞ! 0ð x x xÞ¼ ð x x xÞþ d ð x xÞ ¼ ð x xÞ L t; ; _ ; € L t; ; _ ; € L0 t; ; _ ; € f t; ; _ : L L0 t; ; _ ; (2) dt yields the Euler-Lagrange equation of motion, (10)

124017-2 HIGHER ORDER EQUATIONS OF MOTION AND GRAVITY PHYSICAL REVIEW D 86, 124017 (2012) The expansion of the total time derivative gives that is, all gauge fields are totally symmetric. These fields transform under the generalized gauge transformations as 0ð x x xÞ¼ ð x x xÞþ ð x xÞ L t; ; _ ; € L0 t; ; _ ; € @tf t; ; _ þ x r ð x xÞþx r ð x xÞ q ðt; xÞ!q 0 ðt; xÞ _ f t; ; _ € x_ f t; ; _ : (11) n i1...in n i1...in ¼ q ðt; xÞ@ f ðt; xÞ; The question now is how to employ the gauge principle. n i1...in t i1...in ð x xÞ ð x xÞ q A ðt; xÞ!q A0 ðt; xÞ If we replace, e.g., @tf t; ; _ by a function t; ; _ then n ii1...in n ii1...in this function cannot describe an external field since it ¼ q A ðt; xÞþ@ð f Þðt; xÞ; would depend on the velocity. A given external field should n ii1...in i i1...in q c ðt; xÞ!q c 0 ðt; xÞ be given per se and should not depend on the state of n i1...in n i1...in motion of a particle. The properties of the external field ¼ q c ðt; xÞþnf ðt; xÞ: (20) should be independent of whether the particle is moving n i1...in i1...in through it or not. ¼ One way to introduce functions depending on time and For n 0 they reduce to the ordinary Maxwell gauge position only is to assume that the function fðt; x; x_ Þ is transformations. polynomial in the velocity. That means for some finite As a consequence we obtain the Lagrangian of a particle number N, coupled to the new interaction gauge fields:

XN XN 0 ð x x_ Þ¼ ð xÞ i1 in L ðt; x; x_ ; x€Þ¼L ðt; x; x_ ; x€Þ q ðt; xÞx_ i1 x_ in f t; ; fi1...in t; x_ x_ : (12) 0 n i1...in n¼0 n¼0 XN In this case we can regard the functions f ðt; xÞ¼ i i i i1...in þ q A ðt; xÞx_ x_ 1 x_ n ð xÞ n ii1...in fði1...inÞ t; as gauge functions of an externally given in- n¼0 teraction. For this setting we obtain for the new Lagrange XN function the same equations of motion þ c ð xÞ i1 i2 in qn i1...in t; x€ x_ x_ : (21) n¼1 XN 0 L ðt; x; x_ ; x€Þ¼L ðt; x; x_ ; x€Þþ @ f ðt; xÞx_ i1 x_ in 0 t i1...in The last term can be rewritten as, modulo a total time ¼ n 0 derivative, XN þ @ f ðt; xÞx_ ix_ i1 x_ in i i1...in XN ¼ 1 @ n 0 c ð xÞ i1 i2 in qn i1...in t; x_ x_ x_ XN n¼1 n @t þ nf ðt; xÞx€i1 x_ i2 x_ in : (13) i1...in XN ¼ 1 n 1 c ð xÞ i i1 i2 in qn@i i1...in t; x_ x_ x_ x_ ; (22) ¼ n The gauge principle now allows one to replace these gauge n 1 functions by the gauge fields, so that we effectively have the Lagrangian

@ f ðt; xÞ!q ðt; xÞ; (14) 0 t i1...in n i1...in ð x x xÞ¼ ð x x xÞ L t; ; _ ; € L0 t; ; _ ; € XN ð xÞ! ð xÞ q ~ ðt; xÞx_ i1 x_ i2 x_ in @ifi1...in t; qnAii1...in t; ; (15) n i1...in n¼0 XN ð xÞ! c ð xÞ þ q A~ ðt; xÞx_ ix_ i1 x_ i2 x_ in ; (23) nfi1...in t; qn i1...in t; ; (16) n ii1...in n¼0 where the qn are the coupling parameters to these nth rank potentials. The symmetries of these gauge fields are where

ð xÞ¼ ð xÞ 1 i ...i t; ði ...i Þ t; ; (17) A~ ðt; xÞ :¼ A ðt; xÞ @ð c Þðt; xÞ; (24) 1 n 1 n ii1...in ii1...in n i i1...in

A ðt; xÞ¼Að Þðt; xÞ; (18) ii1...in ii1...in 1 @ ~ ðt; xÞ :¼ ðt; xÞþ c ðt; xÞ: (25) i1...in i1...in n @t i1...in

c ðt; xÞ¼c ð Þðt; xÞ; (19) i1...in i1...in The equations of motion read

124017-3 CLAUS LA¨ MMERZAHL AND PATRICIA RADEMAKER PHYSICAL REVIEW D 86, 124017 (2012)

2 XN d @L0 d @L0 0 ¼ @ L þ q ð@ ~ ðt;xÞn@ ~ ðt;xÞð@ A~ ðt;xÞðn þ 1Þ@ A~ ðt;xÞÞx_ iÞx_ i1 ...x_ in j 0 j 2 j n j i1...in i1 ji2...in j ii1...in i ji1...in dt @x_ dt @x€ n¼0 XN @ XN @ þ ~ ð xÞ i2 in ð þ Þ ~ ð xÞ i1 in qnn ji2...in t; x_ ...x_ n 1 qn Aji1...in t; x_ ...x_ n¼0 @t n¼0 @t XN XN þ ð Þ ~ ð xÞ i2 i3 in ð þ Þ ~ ð xÞ i1 i2 in qnn n 1 ji2...in t; x€ x_ ...x_ n 1 nqnAji1...in t; x€ x_ ...x_ : (26) n¼0 n¼0

The additional gauge interaction adds a time derivative of If the equations of motion are fulfilled, and if the is second order. The principal part of the invariant under the variations (27) and (28), then we obtain remains unaffected. the conserved quantity Below we use the most simple second order Lagrangian d without interaction p ðx x_ Þþðp p_ Þðx x_ ÞþL 2 dt 1 2 m L ðt; x; x_ ; x€Þ¼ x_ 2 x€2; (27) ¼ const; (32) 0 2 2 which leads to a fourth order equation of motion. The where we defined the momenta 2 parameter has the dimension mass time . If one p ¼ r p ¼ r assumes that this additional term has emerged from influ- 1 x_ L and 2 x€L: (33) ences of quantum gravity then it might be natural to identify If the action does not vanish but, instead, changes with a 2 95 2 MPlTPl 10 kg s which is extremely small (the total time derivative of a function FðxðtÞ; x_ ðtÞ;tÞ, then the product of the mass and the square of the Compton time equations of motion in terms of the Euler-Lagrange equa- 71 2 gives for an electron 10 kg s , for a proton we get tions do not change and we have a modified conserved 74 2 77 2 10 kg s , and for a typical atom 10 kg s ). quantity,

III. NOETHER THEOREM p d ð x xÞþðp p Þð x xÞþ 2 _ 1 _ 2 _ L F Also for higher order Lagrangians conservation laws can dt be obtained from the variational principle if we allow ¼ const: (34) nonvanishing variations at the initial and final points. The variations are as usual, From this general Noether theorem we derive the following conserved quantities. t ¼ t þ ðtÞ; (28) A. conservation xðtÞ¼xðtÞþxðtÞ: (29) At first we consider the transformations ðtÞ¼0 and One should bear in mind that here the x need not be xðtÞ¼const. We obtain the conserved momentum variables of the configuration space. For these general P ¼ p p ¼ variations, the variation of the action is 1 _ 2 const: (35) S ¼ S S For the Lagrangian (26) we get Z t ¼ 2 ðxð Þ x_ ð Þ x€ð Þ Þ ::: L t ; t ; t ; t dt P ¼ mx_ þ x ¼ const: (36) t 1Z t2 LðxðtÞ; x_ ðtÞ; x€ðtÞ;tÞdt: (30) This may also directly be inferred from the Euler-Lagrange t1 equations of motion (9). Proceeding as in the first order case described in textbooks we arrive at B. Energy conservation Z 2 t2 d d ð Þ¼ S ¼ rL r L þ r L xdt Next we consider the transformations t 0 and dt x_ dt2 x€ xðtÞ¼0. The corresponding conserved energy is t1 d þ r L ðx x_ Þ E ¼ p x€ þðp p_ Þx_ L ¼ p x€ þ P x_ L x€ dt 1 2 2 ¼ const: (37) d t2 þ rx_ L rx€L ðx x_ ÞþL : (31) dt t1 For the Lagrangian (26) we obtain

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1 1 ::: B. Particle motion E ¼ mx_ 2 þ ð2x x_ x€2Þ: (38) 2 2 The first two time integrations are easily performed and give q C. conservation x þ x ¼ E ð Þ2 þ að Þþb € m 0 t t0 t t0 ; (46) In the next example we assume that the action is invari- 2 ant under the transformations ðtÞ¼0 and x ¼ ’ x. where a and b are integration constants. Next we introduce This corresponds to a and the corresponding con- x x ¼ x þ x x ¼ a new function through 0 with 0 served angular momentum is q E ð Þ2 þ 1 að Þþ1 b 2m 0 t t0 m t t0 m which represents the L ¼ x ðp p Þþx p ¼ x P þ x p solution of the corresponding equation of motion without 1 _ 2 _ 2 _ 2 the fourth order term. Here we assume that is very ¼ const: (39) small, since no deviation from Newton’s second law has x For the Lagrangian (26) we obtain the conserved angular been observed, so far. The differential equation for then momentum reads ::: m q L ¼ x ðmx_ Þþðx x x_ x€Þ¼x P x_ x€ x€ þ x ¼ E : (47) m 0 ¼ const: (40) x ¼ x þ q E A further substitution ^ m m 0 yields the equation for a particle in a harmonic potential, D. Proper Galileo transformation m x€^ þ x^ ¼ 0; (48) At last we consider the Galileo transformations ðtÞ¼0 and x ¼ vt for the Lagrangian (26), where v is 1 x2 which, according to the sign of m , possesses the solution assumed to be very small. The term 2 m _ changes by a total differential of mx v so that we obtain the con- m x^ ¼ A cosð!tÞþB sinð!tÞ for > 0; (49) served quantity ¼ p þðp p Þ x ¼ C 2 _ 2 t m const (41) m x^ ¼ A coshð!tÞþA sinhð!tÞ for < 0; (50) from which we deduce a uniform motion 1 2 p p p 2 _ 2 A B A A x ¼ þ t þ x0: (42) for somepffiffiffiffiffiffiffiffiffiffi , , 1, and 2 and with m m ! ¼ m=. As a consequence, for m= > 0 we arrived at the solution IV. THE MOST SIMPLE GAUGE MODEL, N ¼ 0 q 1 1 A. Equation of motion xðtÞ¼ E ðt t Þ2 þ aðt t Þþ b 2m 0 0 m 0 m The most simple case with nontrivial dynamics is given q for the special case that f in (12) is a function of t and x þ A cosð!tÞþB sinð!tÞ E ; (51) m m 0 only, which means N ¼ 0. Then we have that turns to the standard solution in the limit ! 0. L0ðt;x;x_ ;x€Þ¼L ðt;x;x_ ;x€Þqðt;xÞþqx_ iA ðt;xÞ: (43) 0 i For m= < 0 the solutions become exponentially grow- ing runaway solutions which are in contradiction to For L0 from (26) the equations of motion read :::: physical observations. Accordingly, we have to choose x þ mx€ ¼ qEðt; xÞþqx_ Bðt; xÞ; (44) m= > 0. Since, by convention, m>0 we then have 0. (In principle it is possible to choose the inertial where E and B are the electric and magnetic field derived mass to be negative by definition. This then may induce as usual from the scalar and vector potentials and A. modified sign conventions for coupling constants.) This equation of motion is the standard one with a small The velocity is additional fourth order term. For a fist discussion we may simplify further by taking a q 1 pffiffiffiffiffiffiffi x_ ðtÞ¼ E ðt t Þþ a mA sinð!tÞ vanishing magnetic field, B ¼ 0 and a constant electric m 0 0 m EðxÞ¼E ¼ pffiffiffiffiffiffiffi field 0 const, þ mB cosð!tÞ; (52) :::: x þ mx€ ¼ qE : (45) 0 which also approaches the standard expression for ! 0. This is the equation we would like to solve. The acceleration

124017-5 CLAUS LA¨ MMERZAHL AND PATRICIA RADEMAKER PHYSICAL REVIEW D 86, 124017 (2012) q x€ðtÞ¼ E mA cosð!tÞmB sinð!tÞ; (53) which also can be obtained by multiplying the equation of m 0 motion by x_ and partial integration. Insertion of the solu- xð Þ however, has a large fluctuating term of order 0 which does tion t verifies the constancy of this expression. The not disappear for ! 0. energy is not definite. For very small positive the additional term in the path (50) is a very small but fast oscillating zitterbewegung. One V. THE NEXT GAUGE MODEL, N ¼ 1 may turn the above result into a positive statement: At least A. The equation of motion within a Lagrangian approach and assuming that higher order terms are small, the paths originating from a corre- Now we would now like to explore the model where f is sponding higher order modification are rather inert against a function which is polynomial of first order in the veloc- ð x xÞ¼ ð0Þð xÞþ ð1Þð xÞ i these modifications. The mean orbits behave like orbits ities, that is, f t; ; _ f t; fi t; x_ . Then the given by second order equations of motion. We will present corresponding gauged Lagrange function reads some ways to experimentally search for these fundamental ð x x xÞ¼ ð x x xÞ þ i in Sec. VI. L t; ; _ ; € L0 t; ; _ ; € q0 q0x_ Ai i þ i j þ i c This result may be extended to the case of slowly vary- q1x_ i q1x_ x_ Aij q1x€ i; (60) ing arbitrary electromagnetic fields. The equation of mo- x tion for an arbitrary electromagnetic field (43) may be where all functions depend on t and . Here q0 is the x ¼ x þ x x ð0Þ attacked through the substitution 0, where 0 coupling related to f , and q1 the coupling related to fi. is assumed to solve the equation of motion without the Beside the usual scalar and vector potential, and Ai we c fourth order term. If we assume that the force FðxÞ¼ have in addition two vector potentials i and i and a qEðxÞþqv BðxÞ is very smooth and that the deviation tensorial potential Aij. The gauge transformations are x x rF x is very small (that is, if m €0 and can be given by neglected), then we obtain 0 ¼ ð0Þ 0 ¼ þ ð0Þ q0 q0 @tf ;q0Ai q0Ai @if ; :::: :::: x þ x þ x€ ¼ ð Þ ð Þ 0 m 0: (54) 0 ¼ 1 c 0 ¼ c þ 1 q1i q1 @tfi ;q1 i q1 i fi ; (61) This can be integrated twice, 0 ¼ þ ð1Þ q1Aij q1Aij @ðifjÞ : x€ þ x€ þ mx ¼ at þ b; (55) 0 The corresponding Euler-Lagrange equation is a b where and are two integration constants. Inserting the @L d @L d2 @L equation for x€ yields 0 ¼ 0 0 þ 0 (62) 0 @xi dt @x_ i dt2 @x€i m x€ þ x ¼ 1 Fðx Þþ1 a þ 1 b 0 t : (56) þ þ 2 c m q0@i q0@tAi q1@ti q1@t i (63) x ¼ x 1 a þ 1 b 1 Fðx Þ With a new variable ^ t 2 0 we have m m m þ x_ jð@ ðq A q Þ@ ðq A q Þ m i 0 j 1 j j 0 i 1 i x€^ þ x^ ¼ 0: (57) þ c Þ 2q1@t@½j i (64) x Then again ^ is a fast oscillating term which adds to the 1 j ~ þ ð ~ þ ~ ~ Þ j k standard solution. The total solution then is 2q1 x€ Aij @kAij @jAik @iAjk x_ x_ 2 1 1 1 ~ j xðtÞ¼x ðtÞþ x^ðtÞþ at bþ Fðx ðtÞÞ : (58) 2q1@tAijx_ ; (65) 0 m m m2 0 ~ ¼ c x ð Þ where we defined Aij : Aij @ði jÞ. The first line (61)is This solution consists of the standard solution 0 t which is the main motion and additional small terms: a small fast the (still unspecified) equation of motion of the free parti- oscillating term, a kind of zitterbewegung and a small cle, the second line (62) describes a force due to Eð0Þ ¼r A Eð1Þ ¼ position-dependent , a small offset term, and a two electric fields : @t and : þ 2 c small linearly growing term. In order to be compatible with @t @t , the third line (63) describes the standard ð0Þ observations we choose a ¼ 0 and b ¼ 0 so that the last two magnetic field B :¼ r A as well as an additional Bð1Þ ¼ r ð þ c Þ mentioned terms disappear. magnetic field : @t , hence these last two lines together look like a generalized Lorentz C. Conserved energy force, where the fields are all gauge invariant. The fourth line (64) resembles the form of with a The conserved energy in this most simple model reads ~ connection based on a second rank tensor Aij. m ::: E ¼ x_ 2 þ ð2x x_ x€2Þþq; (59) For the Lagrangian L0 from (26) we obtain the fourth 2 2 order equation of motion,

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::::j j j x ij þ mgijðDx_ x_ Þ þ m@tgijx_ C. Relativistic formulation ð0Þ ð1Þ ð Þ ð Þ x€i c ¼x_ i d c ¼x_ i@ c ¼ q E þ q E þðx_ ðq B 0 þ q B 1 ÞÞ ; (66) It is clear from i dt i t i 0 i 1 i 0 1 i j j x_ x_ @j c i which holds modulo a total time derivative, that i þ i j þ i c where we introduced an effective 3-metric q1x_ i q1x_ x_ Aij q1x€ i q ¼q x_ ið þ @ c Þþq x_ ix_ jðA @ c Þ: (70) g ¼ þ 2 1 A~ ; (67) 1 i t i 1 ij j i ij ij m ij This explains the definitions of Eð1Þ and Bð1Þ. and where the covariant derivative D is formulated with a Furthermore, with xa ¼ðt; xiÞ, a ¼ 0; ...; 3, and the x_ þ Christoffel symbol based on gij. This metric is invariant 3 1 splitting under the gauge transformations (60). The second and third a þ a b ¼ þ þð þ Þ i q0Aax_ q1Aabx_ x_ q0A0 q1A00 q0Ai 2q1Ai0 x_ term on the left-hand side can be regardedR as equation of i j þ i j motion arising from the variation of gijðt; xÞx_ x_ dt.Ifin q1Aijx_ x_ ; (71) ! (65) we let 0 then only the fourth order derivative term we can reproduce, with redefinitions, the gauge ansatz will vanish; the other terms, in particular those containing (59) above. Therefore, the gauge field part in the the metric, remain. Lagrangian (59) can be written in 4-covariant form: One should note that by means of our second order 0 a a b gauge principle we were able to introduce the gravita- L ðt; x; x_ ;x€Þ¼L0ðt; x; x_ ; x€Þq0Aax_ þ q1Aabx_ x_ : (72) tional interaction in terms of a space metric by means of an ordinary gauge field. Together with this metric gauge It is the last term quadratic in the velocities which, field we also introduced an interaction with an electro- when interpreted as gauge field, requires the second magnetic gauge field. Therefore, by means of one gauge order Lagrangian. procedure we introduced the interaction with a gravita- tional field as well as with an electromagnetic field. By VI. COMPARISON WITH EXPERIMENTS omitting the n ¼ 0 part of this gauge formalism, or ¼ We would like to describe possible experiments which equivalently, by setting q0 0, we have a combined gauge formalism for the metric and an electromagnetic may be sensitive to the higher order modifications in the field. We also can choose Eð1Þ ¼ 0 and Bð1Þ ¼ 0 while equations of motion. We take situations where a charged ð Þ ð Þ particle is placed in a constant electric field, e.g., within a keeping A~ 0, E 0 0, and B 0 0. This latter ij capacitor. In standard theory the charged particle will be choice introduces the electromagnetic field and the space accelerated and the final velocity depends on the metric independently. only, not on the spacing between the plates of the capacitor. The structure of the above equation of motion (65)is This will be different in our model. To make the situation :::: as simple as possible we consider a one-dimensional prob- x þ mx€ ¼ Kðt; x; x_ Þ; (68) lem and take as an initial condition that the particle is at ¼ where Kðt; x; x_ Þ is a polynomial of order two in the veloc- absolute rest at t0 0. We take our general solution (50) (in one dimension and ities which slightly generalizes (43). As a consequence, for ¼ ordinary situations (smooth forces) and small we again with t0 0) obtain some zitterbewegung resulting from the inclusion of ð Þ¼ q 2 þ 1 þ 1 þ ð Þþ ð Þ x t 2m E0t m at m b A cos !t B sin !t higher order derivatives. Again, the main (mean) motion is q described by the second order part of the equation of E ; (73) m m 0 motion. q 1 pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi x_ ðtÞ¼ E t þ a A m sinð!tÞþB m cosð!tÞ; B. The conserved energy m 0 m We assume that all fields do not depend on time and (74) determine the conserved energy. Then again we obtain an expression where the space metric appears at the right q x€ðtÞ¼ E Am cosð!tÞBm sinð!tÞ; (75) place. Applying (36) yields m 0 1 1 ::: 3 3 ¼ i j þ þ ð x x x2Þ ::: m 2 m 2 E mgijx_ x_ q0 2 _ € : (69) xðtÞ¼A sinð!tÞB cosð!tÞ; (76) 2 2 As expected, the effective metric (66) enters the conserved and determine the parameters in terms of the initial con- ::: energy. ditions xð0Þ¼0, x_ ð0Þ¼0, x€ð0Þ¼0, and xð0Þ¼0.We

124017-7 CLAUS LA¨ MMERZAHL AND PATRICIA RADEMAKER PHYSICAL REVIEW D 86, 124017 (2012) 0 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffi11 obtain B ¼ 0, a ¼ 0, A ¼ q E , and b ¼ 0. The solution m2 0 @ x_ 2 @ @ mL2 AA then reads x_ ðLÞ¼x_ 1 þ 0 1 cos ! 0 4m L2 2q q 1 0 sffiffiffiffiffiffiffiffiffiffiffiffi11 ð Þ¼ 2 þ ð ð Þ Þ rffiffiffiffiffiffiffi x t E0 t cos !t 1 ; (77) x_ 2mL2 m 2 m þ 0 sin@! AA; (83) 4m L q and we also have ¼ rffiffiffiffi where we substituted E0 L andqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi also inserted the veloc- q x_ ðtÞ¼ E t sinð!tÞ ; (78) ity of the standard theory x_ ¼ 2q. In the standard m 0 m 0 m theory x_ ðLÞ does not depend on L. Here, by varying L we get oscillations in the velocity due to the sin and cos terms, q x_ 3 x€ðtÞ¼ E ð1 cosð!tÞÞ; (79) and also a small offset 0 . m 0 4m L2 Since is assumed to be small, then a change in L rffiffiffiffi will result in fast oscillations which probably cannot be ::: q m resolved. Therefore, averaging over a small L interval xðtÞ¼ E sinð!tÞ: (80) m 0 yields x_ 2 The appearance of the zitterbewegung is connected with hx_ ðLÞi ¼ x_ 1 þ 0 : (84) 0 4m L2 the external field E0. Free particles do not perform such a motion. Therefore in the mean the velocity after traversing the Now we discuss three different measurements whose capacitor is a bit larger. outcome depend on : (i) the time of flight in an accelera- Rewriting the above result as tor, (ii) measurement of the acceleration by atomic inter- deviation ferometry, (iii) noise in electronic circuits. We always hxðLÞi x x2 calculate first order effects. _ _ 0 ¼ _ 0 2 ; (85) x_ 0 4m L A. Determination of time of flight it is clear that one obtains good estimates for for large In standard theory ( ¼ 0) we have the solution xðtÞ¼ velocities x_ 0, short L and small m. Taking, e.g., an electron q 2 with final energy of 10 MeV, a traversed of 2m E0t . For a charged particle being accelerated through its motion through an accelerator of length L we have L ¼ 1m, an accuracy to measure the relative velocity of L ¼ q t2, where is the potential difference the 1%, and assuming that no effect is observed, then we arrive 2m L 50 2 2 ¼ 2m 2 at an estimate 10 kg s . particle traverses. Then t q L so that t is propor- tional to the spacing L and the final velocity does not B. Interferometry depend on the distance L. This will be different for our higher order theory. Acceleration can be measured directly with, e.g., atomic We determine the time the particle needs to traverse interferometry. This has been proposed first by Borde´ [13] the length L. For that we first have to calculate t from and today’s best performance gives an uncertainty of the 8 2 L ¼ xðtÞ: measured acceleration of x€ 10 m=s [14]. However, while this accuracy is valid for a constant acceleration, in q 1 our case we have a fast varying acceleration. L ¼ E t2 þ ðcosð!tÞ1Þ ; (81) m 0 2 m We use the phase shift 2 ¼ ¼ Að!ÞkxT€ ; (86) which isq affiffiffiffiffiffiffi transcendental equation. For 0 we have t ¼ t ¼ 2mL. Therefore we make the approximative where k is the vector of the laser beam acting as 0 qE0 ansatz t ¼ t þ t , where t t and t is of the order . beam splitter and T is the time between the laser pulses. 0 1 1 0 1 For the acceleration we take (78). Since that part of our Solving for t1 gives the time of flight to first order correc- tion in so that acceleration we are interested in and which we like to detect this way is fluctuating, we have to amend the stan- sffiffiffiffiffiffiffiffiffiffi 0 0 1 1 sffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi dard phase shift for a dc acceleration, ¼ kgT2,bya 2mL qE 2mL t ¼ 0 @cos@! A 1A: (82) transfer function Að!Þ which has been determined in qE0 2mL m qE0 Ref. [15]. As an example, we use as charged particle ionized helium and take typical values for the laser wave- We use this time in order to calculate the velocity at length ¼ 780 nm, a pulse spacing time of T ¼ 100 ms, ð Þ¼ ¼ 10 x t L and obtain and an electric field strength of E0 10 V=m.An

124017-8 HIGHER ORDER EQUATIONS OF MOTION AND GRAVITY PHYSICAL REVIEW D 86, 124017 (2012) experimentally reachable accuracy of the phase measure- obtained in the zeroth order formalism, we obtained ment is ¼ 103 rad. for the first order gauge model the standard space-time We are interested in the largest ! which we are able metric carrying the gravitational interaction in terms of a to detect. With the specifications given we can determine gauge field. that ! from the condition Að!Þ¼1025. This gives Then we discussed physical consequences of equations ! ¼ 1012 Hz which is the maximum frequency whose of motion with an additional small even higher order time effect on the phase shift we are able to detect. If we assume derivative, coupled to smooth and slowly varying gauge that nothing is detected this gives an estimate of fields up to our first order gauge formalism. We solved the m equation of motion for the simplest case and deduced ¼ 1050 kg s2: (87) !2 observational consequences in three experimental situ- ations. Leaving aside runaway solutions which are con- nected with a certain choice of the sign of the additional C. Electronic noise higher derivative term and which are contradicting any Another situation where a kind of zitterbewegung of observation, a small even higher order term influences a charged particle may induce an effect is electronics. the time of flight of accelerated particles, yields an accel- As a most simple model we may assume that the particle eration fluctuation accessible in atomic interferometry, and under consideration is located within a capacitor. An also induces a fundamental noise in electronic devices. of this particle will induce an electronic noise Until now no deviation from Newton’s second law has in the electric circuit. This electronic noise can be been observed. Very rough and preliminary estimates for estimated by the parameter characterizing the higher order term could be ChU2i¼mhx_ 2i; (88) derived. However, a comparison with the Planck scale version of this parameter shows that the experimental where U is the voltage. Using only the oscillating terms in estimates are far away from reaching the corresponding the velocity we obtain quantum gravity scale. 1 q 2 The next step is to implement a higher order theory ChU2i¼ E : (89) for point particles in a relativistic context. A possible 2 m 0 ¼ Lagrangepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi function to start with might be L0 Therefore, a modified dynamics will induce an electronic m x_ ax_ b þ x€ax€b, where the dot is the derivative noise—beside other noise like Nyquist noise or a shot ab ab with respect to some parameter along the path. We would noise. However, due to its different characteristics, it might also like to implement our principle in higher order field be disentangled from the other noise sources. Our noise theory, e.g., for scalar field Lagrange densities of the type should be a fundamental noise not depending on tempera- L ¼ ab@ @ þ abcd@ @ @ @ ture, the finite number of charged particles in the electric a b a c b d . A further circuit, etc. A good cryogenic noise limit is of the order step would be to set up some field equations for the new pffiffiffiffiffiffi gauge fields. A first guess might be to have the usual 1nV= Hz [16] in a wide frequency range, that is 1 nV for Maxwell equations for the A and the Einstein field equa- a measurement of duration 1 s. Taking as granted that no a tions for the g . fundamental noise of this kind has been seen under the ab For gauge models with N>1 one obtains equations of conditions of a molecular vacuum and cryogenic tempera- ture, we may get a first estimate 1068 kg s2, where we motion with an acceleration which is multiplied with coef- ficients which depend on the velocity; see Eq. (25). Also all took a capacitance of 0.48 pF, a distance between the þ capacitor plates of 15 m, a voltage of 1000 V. Taking other terms are polynomials of the velocity of degree N 1. into account a bandwidth of 1 GHz, we get a more realistic Altogether, the equation of motion takes a form reminding estimate 1050 kg s2. one of the equation of motion in Finsler geometry. For a recent discussion of a class of Finsler space- and Solar VII. CONCLUSION AND OUTLOOK system tests, see Ref. [17].

We studied the physics resulting from a hypothetically ACKNOWLEDGMENTS given higher order equation of motion substituting standard Newton’s law. In order (i) to investigate the consequences We would like to thank Ch. Borde´, D. Giulini, E. of such models and (ii) to confront such a model with Go¨klu¨, S. Herrmann, J. Kunz, O. Lechtenfeld, and experiments we employed a gauge principle to B. Mashhoon for discussions. Financial support from these equations to external forces. This procedure we the German Research Foundation, in particular within carried through in the framework of higher order the DFG Research training Group 1620 ‘‘Models of Lagrangian formalism. The resulting gauge fields have a Gravity’’, and the Centre for Quantum Engineering richer structure than in the ordinary first order Lagrange and Space-Time Research QUEST is gratefully formalism. Beside the ordinary gauge fields usually acknowledged.

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