En opy 2015,17, 5888-5902; doi:10.3390/e17085888 OPEN ACCESS
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A icle
Gene alised Complex Geome y in The modynamical
Fluc ua ion Theo y
P. Fe nández de Có doba and J. M. Isid o *
Ins i u o Uni e si a io de Ma emá ica Pu a y Aplicada, Uni e sidad Poli écnica de Valencia,
Valencia 46022, Spain; E-Mail: [email p o ec ed].es
*Au ho o whom co espondence should be add essed; E-Mail: [email p o ec ed].es.
Academic Edi o : Geo ge Ruppeine
Recei ed: 29 May 2015 / Accep ed: 19 Augus 2015 / Published: 20 Augus 2015
Abs ac : We p esen a b ie o e iew o some key concep s in he heo y o gene alized
complex mani olds. This new geome y in e pola es, so o speak, be ween symplec ic
geome y and complex geome y. As such i p o ides an ideal amewo k o analyze
he modynamical luc ua ion heo y in he p esence o g a i a ional ields. To illus a e he
use ulness o gene alized complex geome y, we examine a simpli ied e sion o he Un uh
e ec : he he malising e ec o g a i a ional ields on he Sch oedinge wa e unc ion.
Keywo ds: di e en ial-geome ic echniques; luc ua ion heo y; Un uh e ec
1. In oduc ion
The heo y o he modynamical luc ua ions p o ides a solid link be ween mac oscopic and
mic oscopic physics. Classical luc ua ion heo y [1] o en sheds ligh on coun e in ui i e
quan um-mechanical phenomena, hus helping o b idge he gap be ween he classical wo ld and he
quan um wo ld. Fo example, Heisenbe g’s unce ain y p inciple can be nicely illus a ed eso ing o
he heo y o Gaussian luc ua ions a ound he mal equilib ium [2].
On he o he hand, he heo y o he modynamical luc ua ions can be ecas using he geome ic
language o di e en ial mani olds [3–9]. This eexp ession o a physical discipline in mo e abs ac
ma hema ical language goes a long way beyond a me e ew i ing o he concep s in ol ed. I ende s
he heo y mo e e sa ile, enla ging i s scope. Mo eo e , since he ad en o Eins ein’s gene al ela i i y
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a cen u y ago, (pseudo) Riemannian geome y belongs o he echnical skills ha any physicis has o
mas e (a leas a a wo king le el). This places (pseudo) Riemannian geome y a a an age poin .
In he opposi e di ec ion (i.e., he modynamics as applied o geome y) one should men ion a leas wo
de elopmen s. The i s one is a whole body o knowledge on he he modynamics o black holes [10,11].
Mo e ecen ly, he eexp ession o Eins ein’s ela i i y as a he modynamics [12,13] has had a - eaching
consequences o ou unde s anding o space ime.
He e we would like o epo on ano he ecen de elopmen in geome y wi h implica ions on he
he modynamics o luc ua ions: he heo y o gene alized complex mani olds [14,15].
In ying o unde s and he ho ny ela ionship be ween g a i y and he quan um [16–19] i has
been a gued ha g a i y ac s dissipa i ely on quan um sys ems [20]. Speci ically, in he p esence o
a g a i a ional ield, he mal luc ua ions become indis inguishable om quan um luc ua ions [21–23].
This aises he undamen al ques ion: How is one o ea he mal and quan um luc ua ions on he
same oo ing? Is i al oge he possible? We will see he e ha gene alized complex mani olds p o ide
one iable answe o his ques ion, one ha appea s no o ha e been explo ed ye in he geome ical
app oach o he modynamics.
2. Geome y and Fluc ua ions
2.1. Riemannian Geome y
As a e y elemen a y example, conside a he modynamical sys em in an equilib ium s a e desc ibed
by he ollowing a iables: empe a u e T, p essu e Pand olume V. In he Gaussian app oxima ion,
choosing Tand Vas independen a iables, he p obabili y Wo a luc ua ion ∆T,∆Va ound
equilib ium is gi en by [24]
W=W0exp −CV
2kBT2∆T2+1
2kBT∂P
∂V T
∆V2(1)
The he modynamic inequali ies CV>0and (∂P/∂V )T<0ensu e ha he a gumen o he abo e
exponen ial is nega i e de ini e. This sugges s conside ing he ollowing (posi i e de ini e) Riemannian
me ic on he 2-dimensional mani old coo dina ised by T, V :
ds2:= CV
2kBT2dT2−1
2kBT∂P
∂V T
dV2=: gijdxidxj(2)
The me ic coe icien s gij a e o cou se (T, V )-dependen unc ions. This Riemannian s uc u e encodes
all he ele an in o ma ion. Fo example, he a e age alue h (T, V )io an a bi a y unc ion =
(T, V ),
h (T, V )i=Z−1Z (T, V ) exp −gT T T2−gV V V2√gdTdV(3)
whe e Z:= R√gexp (−gT T T2−gV V V2) dTdV, na u ally in ol es he me ic. The ole o
Riemannian geome y in luc ua ion heo y is well known and has been e iewed a leng h in [8].
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2.2. Symplec ic Geome y
As ou s a ing poin he e we will conside a ce ain he modynamical sys em in equilib ium, in o de
o a i e a a co esponding symplec ic s uc u e.
Again in he Gaussian app oxima ion, he p obabili y Wo a luc ua ion ∆P,∆V,∆T,∆Sis gi en
by [24]
W=W0exp −1
2kBT(−∆P∆V+ ∆T∆S)(4)
Assume an equa ion o s a e F(P, V, T)=0 ha can be sol ed o he empe a u e o ob ain
T=g(P, V ). Fo simplici y le us conside an ideal gas, PV =S0T:
W=W0exp −1
2kB−S0
∆P∆V
PV +∆T∆S
T (5)
I is con enien o de ine he dimensionless a iables
p1:= −ln P
P0, q1:= ln V
V0, p2:= ln T
T0, q2:= S
S0
(6)
whe e P0,V0and T0a e e e ence alues. Then Equa ion (5) becomes
W=W0exp −S0
2kB
(∆p1∆q1+ ∆p2∆q2)(7)
We can ega d q1and q2as coo dina es on a he modynamical con igu a ion space S, wi h p1and p2as
hei conjuga e momen a. Thus he q1,p1,q2,p2a e Da boux coo dina es o he symplec ic o m
ω= dp1∧dq1+ dp2∧dq2(8)
In his way we iden i y ∆p1∆q1+ ∆p2∆q2in Equa ion (7) as he symplec ic a ea o a 2-dimensional
su ace Finduced by he luc ua ion:
∆p1∆q1+ ∆p2∆q2=ZF
(dp1∧dq1+ dp2∧dq2)(9)
Finally subs i u ing Equa ion (9) in o Equa ion (7) we ind
W=W0exp −S0
2kBZF
ω(10)
i.e., he p obabili y o his he mal luc ua ion is p opo ional o he exponen ial o he symplec ic a ea
o he luc ua ion su ace F.
The impo ance o symplec ic s uc u es in classical mechanics is widely ecognized and need ha dly
be ecalled [25]. In ac no jus Riemannian geome y, bu also symplec ic geome y, pe ains o he
ealm o he mal luc ua ions: he i s law o he modynamics endows he he modynamic phase space
wi h a con ac s uc u e, which includes symplec ic geome y as a sub-case [3,4,6,7].
A eal 2n-dimensional mani old Mis symplec ic i he e exis s a closed, non-degene a e, ank 2
an isymme ic enso ield ωij de ined e e ywhe e on M. Le xibe local coo dina es a ound x∈M, so
ω=1
2ωijdxi∧dxjwi h ωji =−ωij. Since he ma ix ωij is nonsingula , an in e se πjk exis s such ha
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ωijπjk =δk
i. The Poisson b acke s o wo unc ions , g a e de ined as { , g}:= πjk∂j ∂kg, and he
in eg abili y condi ion dω= 0 u ns ou o be equi alen o he Jacobi iden i y o hese Poisson b acke s.
In his way he ollowing symplec ic analogue o Equa ion (3) allows one o compu e he a e age
alue h io he unc ion on M:
h i=Z−1ZM
exp (−ω)(11)
Abo e, he exponen ial e−ωis de ined by Taylo expansion, powe s being aken wi h espec o he
wedge p oduc . Then he 2n-dimensionali y o he symplec ic mani old picks ou jus one di e en ial
o m ha can be in eg a ed agains M, namely he 2n- o m (−1)nωn/n!; all o he e ms in he Taylo
expansion gi e a anishing con ibu ion when in eg a ed. The ac o (−1)n/n!has been included in
he no maliza ion Z. As had o be he case, his a e age in ol es he da a conce ning he symplec ic
s uc u e on M.
One can also ega d a symplec ic s uc u e as p o iding an isomo phism om he angen space
TxMin o he co angen space T∗
xMa each x∈M. Speci ically, he angen ec o X=Xi∂iis
mapped in o he 1- o m ω(X) = ξ=ξidxi, wi h ξi=ωij Xj. This iewpoin mo i a es he ollowing
de ini ion (equi alen o he abo e, bu mo e use ul o la e applica ions): a symplec ic s uc u e o e
a2n-dimensional mani old Mis an isomo phism ωxbe ween he angen and he co angen ibe s o e
each poin x∈M,
ωx:TxM−→ T∗
xM(12)
such ha , unde he ope a ion o aking he linea dual (deno ed by an as e isk),
ω∗
x=−ωx,∀x∈M(13)
Mo eo e , he in eg abili y condi ion dω= 0 mus be sa is ied.
2.3. Complex Geome y and Kähle Geome y
In o mally one could say ha he imagina y uni is he hallma k o quan um mechanics. Tha
i = √−1pe ains o he quan um wo ld has been e y in e es ingly a gued ecen ly in [26,27]. Mo e
s anda d a gumen s ha e been known o long; such a e he hea equa ion in imagina y ime i , o he
ac ha quan um commu a o s [·,·] o mally equal √−1 imes classical Poisson b acke s {·,·}. He e
we will b ie ly ecall he ole played by complex s uc u es in he heo y o cohe en s a es [28,29].
Le Mbe a eal 2n-dimensional phase space endowed wi h he symplec ic o m ω. Fo simplici y
le us also assume ha Madmi s a holomo phic a las compa ible wi h he symplec ic s uc u e ( his
compa ibili y condi ion is called he Kähle p ope y). In plain wo ds, he eal and imagina y pa s o he
holomo phic coo dina es zja e Da boux coo dina es o ω(he e assumed dimensionless o simplici y):
zj=1
√2qj+ ipj, j = 1, . . . , n (14)
Upon quan isa ion, he Da boux coo dina es qjand pjbecome ope a o s Qjand Pjon Hilbe space
sa is ying he Heisenbe g algeb a [Qj, Pk] = iδj
k. C ea ion and annihila ion ope a o s a e de ined in he
s anda d ashion: A†
j:= (Qj−iPj)/√2,Aj:= (Qj+ iPj)/√2, and quan um exci a ions a e measu ed
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wi h espec o a acuum s a e |0isa is ying Aj|0i= 0, o all j= 1, . . . , n. Cohe en s a es |zjia e
eigen ec o s o Aj, he eigen alues being he holomo phic coo dina es (14):
Aj|zji=zj|zji, j = 1, . . . , n (15)
(No sum o e jimplied). In o de o illus a e ou poin le us conside a 1-dimensional ha monic
oscilla o . The expec a ion alue o he Hamil onian ope a o H=A†A+ 1/2in he s a e |ziequals
hz|H|zi=|z|2+ 1/2. Since he ene gy luc ua ion in he s a e |ziequals
(∆H)z=|z|, z ∈C(16)
he ela i e luc ua ion goes, o la ge enough |z|, like
(∆H)z
hz|H|zi≃1
|z|,|z| → ∞ (17)
Bu 1/|z|is he in e se o he squa e oo o he Kähle po en ial K(z, ¯z) := |z|2 o he Euclidean
me ic on he complex plane C. This simple example illus a es he impo an ole played by complex
mani olds in he quan um heo y.
E e y complex mani old Madmi s a (posi i e de ini e) He mi ian me ic hijd¯zidzj ha is compa ible
wi h he complex s uc u e [30]. Then an analogue o Equa ions (3) and (11) gi es us he a e age alue
h io a unc ion on M:
h i=Z−1ZM
exp −hij ¯zizj√h
n
Y
k=1
d¯zk∧dzk(18)
The no maliza ion Zincludes all ac o s o i = √−1coming om he olume elemen , and h:=
|de hij|. As had o be he case, his a e age in ol es he da a conce ning he complex s uc u e on M.
Fo mally, a complex s uc u e Jo e a eal 2n-dimensional mani old Mis an endomo phism o he
angen ib e o e each poin x∈M
Jx:TxM−→ TxM(19)
sa is ying
J2
x=−1,∀x∈M(20)
as well as he in eg abili y condi ion ha he Nijenhuis enso N anish iden ically. (We will
no w i e down he Nijenhuis enso explici ly; see e e ence [30] o de ails). Roughly speaking,
Equa ion (20) exp esses he exis ence o he imagina y uni i = √−1locally a ound he poin x∈M.
The in eg abili y condi ion N= 0 ensu es ha he complex coo dina es hus cons uc ed locally uly
ans o m holomo phically ac oss di e en coo dina e pa ches on he mani old M. (The Kähle p ope y
assumed in Equa ion (14) abo e is an addi ional hypo hesis, ha an a bi a y complex mani old may, bu
need no , sa is y in gene al).

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2.4. Gene alized Complex Geome y
Ou o iginal mo i a ion was he s a emen [21–23] ha , in he p esence o a g a i a ional ield,
quan um luc ua ions become indis inguishable om he mal luc ua ions. We ha e a gued ha he mal
luc ua ions a e associa ed wi h symplec ic s uc u es, while quan um luc ua ions come along wi h
complex s uc u es. How, hen, is one o ea he mal and quan um luc ua ions on he same oo ing?
This is i ially achie ed by hose phase spaces M ha quali y as Kähle mani olds. Howe e , he Kähle
condi ion is e y es ic i e: no only does Mha e o be simul aneously complex and symplec ic; hese
wo independen s uc u es also ha e o be compa ible.
In e e ences. [3,4] he geome y o he he modynamic phase space (including luc ua ions) esul s in
a pa a-Sasakian mani old, which is he con ac -geome y equi alen o a Kähle mani old in symplec ic
geome y. This means ha i one es ic s o a p ope e en-dimensional subspace, he geome y is indeed
ha o a Kähle mani old. This geome y achie es he goal o ea ing he mal and quan um luc ua ions
on he same oo ing.
Gene alized complex s uc u es (GCS) also achie e he goal o p o iding a uni ied amewo k o
he mal and quan um luc ua ions. The ollowing is a b ie summa y o GCS ex ac ed om [14], duly
ailo ed o mee ou needs. Fo simplici y we p e e o wo k locally a ound a poin x∈M. Global
issues can be aken ca e o by he co esponding in eg abili y condi ions, o be men ioned along he way
whene e necessa y. Fo ou pu poses he 2n-dimensional mani old Mis assumed o be a phase space,
ha is, M=T∗S, o a ce ain n-dimensional con igu a ion space S.
Ra he han conside ing he ib es TxMo T∗
xMsepa a ely, in gene alized complex geome y one
conside s hei di ec sum: o e each poin x∈Mone e ec s he ib e TxM⊕T∗
xM. The o al space o
he bundle so cons uc ed is 6n-dimensional: 2ndimensions o he base M,4ndimensions o he ib e.
An inne p oduc is de ined on he ib e TxM⊕T∗
xM:
hX+ξ, Y +ηi:= 1
2(ξ(Y) + η(X)) (21)
Abo e, X, Y ∈TxMa e angen ec o s, while ξ, η ∈T∗
xMa e 1- o ms, all e alua ed a x∈M. I u ns
ou ha his inne p oduc is pseudo-Riemann wi h signa u e (2n, 2n). Hence he Lie g oup SO(2n, 2n)
ac s on TxM⊕T∗
xMby isome ies. I is con enien o block-decompose he Lie algeb a so(2n, 2n)as
ollows: A β
B−A∗!(22)
The diagonal blocks Aand A∗a e endomo phisms o hei espec i e (sub) ibe s, A∈End(TxM)and
A∗∈End(T∗
xM), while he o diagonal blocks Band βconnec hese wo (sub) ibe s as pe
B:TxM−→ T∗
xM, β :T∗
xM−→ TxM(23)
Mo eo e , upon aking he dual we ha e B∗=−B,β∗=−β. This an isymme y allows us o ega d
he block Bas a 2- o m in Λ2T∗
xMi we se
B(X) = iXB(24)
Fo illus a i e pu poses le us exp ess Equa ion (24) in local coo dina es xia ound a poin x∈M, so
Bbecomes he ma ix Bij. Gi en he ec o X=Xj∂j∈TxM, he objec iXBis de ined o be he
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co ec o whose componen s a e BijXj∈T∗
xM. We see ha his is exac ly he way a symplec ic o m ω
beha es. Since ωcan be ega ded as an elemen o Λ2T∗
xM, so can B. (Con a y o ω, howe e , Bneed
nei he be closed no non-degene a e).
The pa icula isome ies o he ib e TxM⊕T∗
xMob ained by se ing A= 0 = βin Equa ion (22)
and exponen ia ing,
exp 0 0
B0!= 10
B1!(25)
a e he pseudo-o hogonal ans o ma ions
X+ξ−→ X+ξ+iXB(26)
The isome ies (26), called B- ans o ma ions, will play an impo an ole.
A gene alized complex s uc u e o e M, deno ed J, is an endomo phism o he ib e o e each
x∈M,
Jx:TxM⊕T∗
xM−→ TxM⊕T∗
xM(27)
such ha he ollowing wo condi ions hold. Fi s ,
J2
x=−1,∀x∈M(28)
Second,
J∗
x=−Jx,∀x∈M(29)
The abo e wo condi ions a e o mula ed locally a ound any x∈M; as usual hey need no be
compa ible wi h changes o coo dina e cha s on M. The Cou an in eg abili y condi ion, whose alidi y
we will hence o h assume wi hou s a ing i s con en s explici ly, ensu es his compa ibili y; see [14,15]
o de ails.
Compa ing now Equa ions (29) and (13), we a e led o he pa icula case when Ja x∈Mis
gi en by
Jωx= 0−ω−1
x
ωx0!(30)
whe e ωis a symplec ic o m. One says ha his Jωde ines a GCS o symplec ic ype.
Simila ly, he compa ison o Equa ions (28) and (20) sugges s he pa icula case o a GCS gi en by
JJx= −Jx0
0J∗
x!(31)
whe e Jis a complex s uc u e. We say ha he abo e JJde ines a GCS o complex ype.
Fu he mo e, GCS succeed a in e pola ing be ween he abo e opposi e ypes, he symplec ic ype and
he complex ype; le us explain his mo e ca e ully. A poin x∈Mis said o be egula i i possesses
a neighbo hood Nxon which he e exis s a Poisson s uc u e ω−1wi h cons an ank. In a neighbo hood
Nxo any egula poin x∈Mone can de ine a di eomo phism and a B- ans o ma ion, he combined
ac ion o which maps Nxin o he p oduc Cx×Rx⊂Ck×R2n−2k. He e Cxis an open se wi hin he
s anda d complex mani old Ck, and Rxis an open se wi hin he s anda d symplec ic mani old R2n−2k.
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The nonnega i e in ege kis called he ype o he GCS J, he limi ing cases o Equa ions (30) and (31)
espec i ely co esponding o k= 0 and k=n. As desc ibed in [14,15], he ype kneed no be cons an
ac oss M: i may a y om one poin o ano he in M.
In plain wo ds, any gene alized complex mani old ac o izes locally as he p oduc o a complex
mani old imes a symplec ic mani old.
Finally assume ha Mis a linea space. Then any gene alized complex s uc u e o ype k= 0 is he
B- ans o m o a symplec ic s uc u e. This means ha any gene alized complex s uc u e o ype k= 0
can be w i en as
e−BJωeB= −ω−1B−ω−1
ω+Bω−1B Bω−1!(32)
o a ce ain 2- o m B; use has been made o Equa ions (25) and (30). Simila ly any gene alized complex
s uc u e o ype k=no e a linea mani old Mis he B- ans o m o a complex s uc u e,
e−BJJeB= −J0
BJ +J∗B J∗!(33)
a e using Equa ions (25) and (31). When Mis an a bi a y smoo h mani old, no necessa ily a linea
space, s a emen s (32) and (33) emain basically ue, wi h some mino modi ica ions equi ed; see
e e ences [14,15] o de ails.
The consequences o he abo e become immedia ely appa en . Le us o simplici y assume ha he
ype kis cons an ac oss M. Then any GCS wi h an ex emal alue o k,i.e., ei he k= 0 o k=n, can
always be educed o he co esponding canonical o m (30) o (31) by means o a B- ans o ma ion.
Thus k= 0 co esponds o a he mal desc ip ion o phenomena, while k=nco esponds o a quan um
desc ip ion o phenomena, no in e pola ion exis ing be ween he wo desc ip ions. Nonex emal alues
o he ype, i.e., such ha 06=k6=n, con ain bo h he mal and quan um desc ip ions simul aneously.
A e age alues h io unc ions on gene alized complex mani olds a e de ined by an ob ious
modi ica ion o he p oduc o he igh -hand sides o Equa ions (11) and (18).
3. When “Quan um” Becomes “The mal”
Any g a i a ional ield is locally equi alen o an accele a ed ame. In an accele a ed ame, quan um
becomes he mal; his is basically he con en o he Un uh e ec [31] (in an admi edly lax o mula ion
ha is howe e p ecise enough o ou pu poses). Wi hou using he ull appa a us o ela i is ic
quan um ield heo y, le us see how quan um can become he mal in he simpli ied se up o he quan um
mechanics o a non ela i is ic pa icle. This unde s ood, we will analyse he ole played by he GCS on
phase space unde he passage om an ine ial ame o an accele a ed ame. We will conclude ha he
ans o ma ion law o he Sch oedinge wa e unc ion unde he passage o a nonine ial ame (as in he
Un uh e ec ) is go e ned by a B- ans o ma ion o he GCS on phase space.
A ema k is in o de . The g a i a ional ield conside ed he e mus be weak in o de o ule ou e ec s
such as, e.g., ela i is ic speeds, o he likely b eakdown o s anda d quan um mechanics in he p esence
o e y s ong g a i a ional ields [20]. Such phenomena lie beyond ou scope.
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3.1. Inclusion o a G a i a ional Field
In la Euclidean space R3, le Kdeno e an ine ial ame wi h o igin Oand axes Ox,Oy and Oz.
Le K0deno e a uni o mly accele a ed ame, wi h o igin O0and axes O0x0,O0y0and O0z0 espec i ely
pa allel o Ox,Oy and Oz. Fo simplici y we will assume ha , a = 0, he wo o igins Oand O0
coincide, hei ela i e eloci y also anishing a = 0. Le he accele a ion ~α o K0wi h espec o
Kbe (α, 0,0), wi h αa cons an . Coo dina es (x, y, z)wi h espec o Ka e ela ed o coo dina es
(x0, y0, z0)wi h espec o K0as pe
x=x0+1
2α 2, y =y0, z =z0, = 0(34)
We conside a poin pa icle o mass m ixed o he o igin O0, hus a es wi h espec o K0. I Hdeno es
he Hamil onian o he pa icle as seen om he ine ial ame K, hen he Hamil onian H0in K0 eads
H0=H−pxα +m
2α2 2(35)
wi h he momen a pxand p0
x ela ed as pe p0
x=px−mα . In he ine ial ame Kwe ha e a Sch oedinge
equa ion i¯h∂ψ/∂ =Hψ. Ou aim is o de i e a ans o ma ion law o he wa e unc ion ψsuch ha , in
he accele a ed ame K0, he Sch oedinge equa ion will ead i¯h∂ψ0/∂ =H0ψ0. Fo his pu pose le us
make he Ansa z
ψ0=ψexp [ ( )] (36)
( )being an unde e mined unc ion o he ime a iable. In his way we a i e a he ollowing
di e en ial equa ion o he unknown unc ion :
i¯hd
d =−pxα +1
2mα2 2(37)
D opping an i ele an in eg a ion cons an and subs i u ing he esul in o Equa ion (36) leads o
ψ0= exp −i
¯h1
6mα2 3−1
2pxα 2ψ(38)
Clasically, he pa icle is a es in he ame K0, so p0
x= 0 implies px=mα . Quan um-mechanically
we can only s a e ha he cen e o mass emains a es a x0= 0, he wa epacke sp eading a ound
his a e age posi ion. Wi h his unde s anding we can also se hpxi=px=mα in Equa ion (38). We
conclude ha , aking he wa e unc ion in he accele a ed ame o be
ψ0= exp i
¯h
1
3mα2 3ψ(39)
ensu es he o m in a iance o he Sch oedinge equa ion unde he ans o ma ion om an ine ial
ame o an accele a ed ame. Fo ime lapses ha a e sho enough, and/o o accele a ions ha a e
weak enough, he speeds a ained will ne e become ela i is ic. Wi hin his limi ed ange, New onian
mechanics (and i s quan um coun e pa , he Sch oedinge equa ion) can be us ed.