Finite-Mode Method (FM²): A Rigorous Lorentzian Path Integral Framework and its Implications for Induced Gravity

Fini e-Mode Pa h In eg a ion in Cu ed Space ime:
F om Real-Time S abili y o he Eme gence o Geome y ia Induced
G a i y
Yu-Wei Chen
Depa men o Physics, Na ional Cheng Kung Uni e si y, Tainan 701, Taiwan
[email p o ec ed]
No embe 24, 2025
Abs ac
The Feynman pa h in eg al, while cons i u ing he co ne s one o mode n quan um ield
heo y, su e s om se e e measu e- heo e ic pa hologies in Minkowski space ime, pa icu-
la ly in ime-dependen cosmological backg ounds whe e he adi ional Wick o a ion be-
comes ill-de ined due o he complexi ica ion o he me ic. In his wo k, we p esen he
Fini e-Mode Me hod (FM2), a igo ous cons uc i e o malism o eal- ime pa h in e-
g a ion. By p ojec ing he quan um ield on o a comple e Gabo ame—explici ly de ined
as he p oduc o a Gaussian window and a plane wa e—we egula ize he in ini e-dimensional
in eg al in o a sequence o con e gen ini e-dimensional ma ix in eg als. We sol e he con-
e gence p oblem o oscilla o y F esnel in eg als wi h inde ini e quad a ic o ms by applying
Le sche z Thimble heo y, explici ly demons a ing he con ou o a ions equi ed o bo h
spacelike and imelike modes. We p o e ha he space ime me ic is no a undamen al back-
g ound bu eme ges as he e ec i e e ac i e index o he acuum, igo ously sa is ying he
Hadama d singula i y condi ion. The o malism is alida ed by non-pe u ba i ely de i ing
he exac Mehle ke nel o he quan um ha monic oscilla o ia he Gel’ and-Yaglom
heo em and he (1+1)D Casimi ene gy ia Eule -Maclau in expansion. Finally,
u ilizing he Hea Ke nel expansion o he FM2e ec i e ac ion, we p o ide a i s -p inciples
de i a ion o he Eins ein Field Equa ions, demons a ing ha g a i y eme ges as a s a-
is ical en opic o ce induced by he Gabo - unca ed quan um luc ua ions o ma e ields.
Con en s
1 In oduc ion: The C isis o Wick Ro a ion 3
2 Me hodology: The Fini e-Mode Me hod (FM2)3
2.1 Failu e o he Pu e Fou ie Basis ........................... 3
2.2 The Gabo F ame and Field Expansion ....................... 4
2.3 F om he Ac ion o a Fini e-Dimensional Quad a ic Fo m ............. 5
2.4 Le sche z Thimble Regula iza ion o he F esnel In eg al .............. 6
2.5 Recons uc ion o he Full Pa h In eg al ....................... 7
3 Op ical Analogy: E ec i e Me ic as Re ac i e Index 7
1
4 Ve i ica ion I: The Quan um Ha monic Oscilla o 7
4.1 The Gel’ and–Yaglom Theo em ............................ 8
4.2 FM2De e minan Limi and Mehle Ke nel ..................... 8
5 Ve i ica ion II: (1+1)D Casimi E ec om Ma ix T ace 9
5.1 FM2Ma ix Cons uc ion and Ene gy De ini ion .................. 9
5.2 Eule –Maclau in Expansion .............................. 9
6 Eme gence o G a i y: Induced Eins ein Field Equa ions 10
6.1 S ep 1: Ma e Ac ion wi hou Ba e G a i y .................... 10
6.2 S ep 2: Schwinge P ope Time and Hea Ke nel .................. 10
6.3 S ep 3: Va ia ional P inciple and Eins ein Equa ions ................ 11
7 Conclusion 12
2
1 In oduc ion: The C isis o Wick Ro a ion
The Feynman pa h in eg al o mally de ines he ansi ion ampli ude as a sum o e all
possible ield con igu a ions:
Z=ZDϕ e i
ℏS[ϕ].(1)
In s anda d Quan um Field Theo y (QFT), ma hema ical igo is ypically sough ia Wick
o a ion o Euclidean ime ( → −iτ). This elies on he assump ion ha he Euclidean ac ion
SEis posi i e de ini e, allowing he oscilla o y ac o eiS o become a Bol zmann weigh e−SE.
Howe e , in Quan um Cosmology, his p ocedu e is undamen ally lawed. Conside a
F iedmann-Robe son-Walke (FRW) uni e se wi h a scale ac o a( ) = 1 + ϵ . Unde Wick
o a ion, he me ic de e minan becomes complex:
√−g∝a( )3 →−iτ
−−−−→ (1 −iϵτ)3.(2)
The eal pa o he Euclidean measu e ac o , Re[(1 −iϵτ)3]=1−3ϵ2τ2, becomes nega i e o
Euclidean imes τ > 1/(√3ϵ). Consequen ly, he ”Euclidean” kine ic e m lips sign, ende ing
he ac ion unbounded om below and causing he pa h in eg al o di e ge ca as ophically. To
esol e his, a s ic ly eal- ime o malism is equi ed.
2 Me hodology: The Fini e-Mode Me hod (FM2)
The cen al idea o FM2is o ebuild he Feynman pa h in eg al as he limi o a sequence o
ini e-dimensional oscilla o y in eg als. Ins ead o s a ing om a Euclidean unc ional measu e
and analy ically con inuing back o eal ime, we emain s ic ly on he Lo en zian axis and
egula ize he pa h in eg al by p ojec ing he ield on o a ca e ully chosen ime– equency ame.
In his sec ion we i s explain why he s anda d plane-wa e (Fou ie ) basis is ma hema ically
ill-sui ed o eal- ime cu ed backg ounds, hen cons uc he Gabo ame, de i e he ini e-
mode quad a ic ac ion, and inally show how he esul ing F esnel in eg als a e made con e gen
ia Le sche z himbles.
2.1 Failu e o he Pu e Fou ie Basis
Le us deno e space ime coo dina es by x= ( , x) and conside a eal scala ield ϕ(x) on a
ime-dependen backg ound such as an FRW space ime. Fo mally, one may a emp o expand
ϕ( , x) = Zdω d3k
(2π)4˜
ϕ(ω, k)ei(ω −k·x),(3)
and ew i e he ac ion in e ms o he Fou ie coe icien s ˜
ϕ(ω, k). Howe e , his pu ely Fou ie -
based desc ip ion su e s om h ee s uc u al p oblems on a ime-dependen backg ound gµν( , x):
1. Non- anishing bounda y e ms. Plane wa es eiω do no decay as | | → ∞. When
de i ing he quad a ic o m o he ac ion, in eg a ions by pa s in gene ically p oduce
bounda y e ms o he o m hap( )¯
( )∂ g( )i+∞
=−∞,(4)
whe e a( ) is he scale ac o , pis a powe de e mined by he dimension, and , g a e mode
unc ions. Wi h plane wa es his e m does no anish and has o be disca ded by hand
(e.g., “swi ching o ” he in e ac ion in he dis an pas / u u e o enclosing he uni e se
in a box), which is concep ually and ma hema ically unsa is ac o y.
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2. Lack o ime localiza ion. Fou ie modes a e global in ime: a single mode eiω ex ends
ac oss he en i e cosmic his o y. In an FRW uni e se, he scale ac o a( ) encodes s ong
ime dependence nea he Big Bang and in la e- ime accele a ion. A global basis canno
isola e he local geome ic ea u es o a gi en epoch, making i di icul o con ol he
sho -dis ance / sho - ime s uc u e o he p opaga o in a co a ian way.
3. Dense ac ion ma ix. Fo a non-s a iona y backg ound, di e en Fou ie modes a e
s ongly coupled. The quad a ic o m o he ac ion in Fou ie space,
S[˜
ϕ] = 1
2Zdω dω′d3k d3k′˜
ϕ(ω, k)∗A(ω, k;ω′,k′)˜
ϕ(ω′,k′),(5)
in ol es a ke nel A ha is gene ically dense. This makes he spec al analysis o he
ope a o ˆ
Kand he e alua ion o de Aex emely cumbe some, bo h concep ually and
nume ically.
These sho comings mo i a e he in oduc ion o a ime– equency localized ame ha decays
in and yields a spa se, s uc u ed ac ion ma ix. This is p ecisely he ole played by he Gabo
ame in FM2.
2.2 The Gabo F ame and Field Expansion
We de ine he (1+1)-dimensional Gabo a om as
gj( , x) = g(m,n,k)( , x) = exp−( − m)2
2σ2
| {z }
Gaussian window
eiωn
|{z}
ime modula ion
eikx
|{z}
spa ial ca ie
,(6)
whe e mis he ime cen e , ωn he cen al equency, k he spa ial momen um, and σis he
ime–window wid h. Fo de ini eness we i s wo k in (1+1) dimensions; gene aliza ion o highe
dimensions is s aigh o wa d.
The key p ope ies o his ame a e:
•Time– equency localiza ion: each gjis localized a ound ( m, ωn) wi h Gaussian decay in
.
•App oxima e comple eness: he collec ion {gj}j∈Z3 o ms a ( edundan ) Gabo ame o
L2(R2); any squa e-in eg able ield can be expanded as
ϕ( , x) = lim
N→∞
N
X
j=−N
ujgj( , x), uj∈C,(7)
wi h con e gence in he L2no m.
•Au oma ic anishing o bounda y e ms: because o he Gaussian window, all mode unc-
ions and hei de i a i es anish exponen ially as | | → ∞, so ha su ace e ms om
in eg a ion by pa s anish iden ically:
[ap( ) ¯gm( , x)∂ gn( , x)]+∞
=−∞ = 0.(8)
In p ac ice, FM2p oceeds by unca ing he expansion o a ini e numbe No Gabo modes,
ϕN( , x) =
N
X
j=1
ujgj( , x),(9)
and hen aking N→ ∞ a he end o he calcula ion. This is he o igin o he e m ini e-mode.
4
2.3 F om he Ac ion o a Fini e-Dimensional Quad a ic Fo m
To make he cons uc ion conc e e, le us wo k wi h a eal scala ield on a (1+1)-dimensional
FRW backg ound,
ds2=−d 2+a2( )dx2,√−g=a( ),(10)
wi h ac ion
S[ϕ] = 1
2Zd dx a( )−(∂ ϕ)2+1
a2( )(∂xϕ)2−m2ϕ2.(11)
Subs i u ing he ini e-mode expansion ϕN( , x) in o (11) and using linea i y, we ob ain
SN(u) = 1
2
N
X
m,n=1
¯um(AN)mn un=1
2u†ANu,(12)
whe e he ma ix elemen s a e
(AN)mn =Zd dx a( )−(∂ gm) (∂ gn) + 1
a2( )(∂xgm) (∂xgn)−m2gmgn.(13)
Gi en he explici o m (6), he de i a i es a e
∂ gj( , x) = − − m
σ2+iωngj( , x)≡D ,j( )gj( , x),(14)
∂xgj( , x) = ik gj( , x).(15)
Inse ing hese, he ma ix na u ally decomposes in o h ee physical blocks:
(AN)mn =Kmn +Gmn +Mmn,(16)
wi h
Kmn =−Zd dx a( )D ,m( )D ,n( ) ¯gmgn,(kine ic block),(17)
Gmn =Zd dx a( )
a2( )(ikm)(ikn) ¯gmgn=−Zd dx 1
a( )kmkn¯gmgn,(g adien block),(18)
Mmn =−Zd dx a( )m2¯gmgn,(mass block).(19)
Because o he Gaussian ime window and he oscilla o y spa ial ac o , all hese in eg als a e
con e gen o each ini e N.
A his s age, he con inuum pa h in eg al
Z=ZDϕ e i
ℏS[ϕ](20)
has been igo ously educed o a ini e-dimensional F esnel in eg al o e he complex coe icien s
{uj}:
ZN=ZR2N
dN(Re u)dN(Im u) expi
2ℏu†ANu.(21)
The FM2p esc ip ion is o e alua e ZNexac ly o each ini e Nand hen s udy he limi
N→ ∞.
5

2.4 Le sche z Thimble Regula iza ion o he F esnel In eg al
The in eg al (21) is highly oscilla o y because ANinhe i s he Lo en zian signa u e o he
unde lying space ime: i has bo h posi i e and nega i e eigen alues. Nai ely applying he la -
space Gaussian o mula would be ill-de ined. FM2o e comes his by applying Pica d–Le sche z
heo y mode by mode.
Le λkand (k)be he eigen alues and eigen ec o s o AN, so ha in he eigenbasis he
quad a ic o m is diagonal,
SN(u) = 1
2
N
X
k=1
λk|yk|2, yk= ( (k))†u.(22)
The in eg al ac o izes:
ZN=
N
Y
k=1 ZR2
d2ykexpi
2ℏλk|yk|2.(23)
Each ac o is a wo-dimensional F esnel in eg al. Fo a gi en eigen alue λkwe de o m he
in eg a ion con ou in o he complex plane along he s eepes descen di ec ion ( he Le sche z
himble):
•I λk>0 (“ma e -like” mode), we o a e he con ou by +45◦:
yk=eiπ/4xk, xk∈R,
so ha i
2λky2
k→ −1
2λkx2
k,
and he in eg al becomes a con e gen eal Gaussian.
•I λk<0 (“g a i y / con o mal” mode), we o a e by −45◦:
yk=e−iπ/4xk, xk∈R,
leading again o a nega i e-de ini e quad a ic o m in xk.
In bo h cases he in eg al yields
ZJk
d2ykexpi
2ℏλk|yk|2=2πiℏ
λk
,(24)
up o an unimpo an o e all phase common o all modes. Taking he p oduc o e kand using
Qkλk= de AN, we ob ain he exac closed- o m exp ession
ZN=NNs(2πiℏ)N
de AN
,(25)
whe e NNis a phase ac o independen o he backg ound geome y.
6
2.5 Recons uc ion o he Full Pa h In eg al
Finally, he FM2de ini ion o he ull Lo en zian pa h in eg al is
Z[g] = lim
N→∞ZN[g] = lim
N→∞NNs(2πiℏ)N
de AN[g].(26)
He e AN[g] is he ini e-mode Gale kin p ojec ion o he co a ian kine ic ope a o
ˆ
Kg=−□g+m2,(27)
on o he Gabo ame adap ed o he FRW backg ound. The e ec i e ac ion is hen
Γe [g] = −iln Z[g] = i
2lim
N→∞ T ln AN[g] + cons .,(28)
which is he objec we analyze in la e sec ions ia hea -ke nel and spec al echniques o de i e
he Casimi ene gy, he Mehle ke nel, and ul ima ely he induced Eins ein–Hilbe e m.
This comple es he ini e-mode econs uc ion o he eal- ime Feynman pa h in eg al in a
ime-dependen FRW backg ound.
3 Op ical Analogy: E ec i e Me ic as Re ac i e Index
We now demons a e ha he space ime me ic can be econs uc ed om he singula i y
s uc u e o he p opaga o de i ed ia FM2.
In he high- equency limi (N→ ∞), he p opaga o Gis he in e se o he kine ic ope a o .
In Fou ie space (ω, k), u ilizing he coe icien s om he ANma ix cons uc ion in an FRW
backg ound,
˜
G(ω, k)∼1
−a3( )ω2+a( )k2=1
a( )−a2( )ω2+k2.(29)
The pole s uc u e de ines he dispe sion ela ion
a2( )ω2=k2,(30)
implying a phase eloci y p=ω/|k|= 1/a( ). Viewing he acuum as an op ical medium, he
e ec i e e ac i e index is
n( ) = c
p
=a( ).(31)
Pe o ming he Fou ie ans o m back o posi ion space in 3 + 1 dimensions,
G(x, x′)∼Zeik·(x−x′)
k2d4k∼1
σ2(x, x′),(32)
whe e σ2(x, x′)≃ −a2( )∆ 2+∆x2is he squa ed geodesic dis ance de ined by he FRW me ic.
This 1/σ2beha io p ecisely ma ches he Hadama d singula i y condi ion equi ed o QFT in
cu ed space ime. Thus, FM2 econs uc s he ligh -cone s uc u e o Gene al Rela i i y om
he p incipal symbol o he kine ic ope a o .
4 Ve i ica ion I: The Quan um Ha monic Oscilla o
We now use FM2 o de i e he p opaga o o he ha monic oscilla o and e i y ha he
de e minan limi ep oduces he exac Mehle ke nel.
7
4.1 The Gel’ and–Yaglom Theo em
Conside he in e ac ing ope a o
ˆ
OV=−∂2
−ω2(33)
and he ee ope a o ˆ
O0=−∂2
,(34)
bo h de ined on pa hs x( ) wi h ixed endpoin s a = 0 and =T. The Gel’ and–Yaglom
heo em s a es ha he a io o unc ional de e minan s is gi en by
lim
N→∞
de ˆ
OV
de ˆ
O0
=ψV(T)
ψ0(T),(35)
whe e ψ( ) sa is ies he homogeneous equa ion
ˆ
Oψ( ) = 0, ψ(0) = 0,˙
ψ(0) = 1.(36)
4.2 FM2De e minan Limi and Mehle Ke nel
In he FM2 o malism, we p ojec he ope a o on o a ini e Gabo subspace:
A(V)
N=PNˆ
OVPN, A(0)
N=PNˆ
O0PN,(37)
so ha he ini e-mode pa i ion unc ions ead
Z(V)
N∝1
qde A(V)
N
, Z(0)
N∝1
qde A(0)
N
.(38)
Taking N→ ∞ and using he Gel’ and–Yaglom heo em,
de A(V)
N
de A(0)
N−−−−→
N→∞
de ˆ
OV
de ˆ
O0
=ψV(T)
ψ0(T).(39)
Fo he ha monic oscilla o ,
¨
ψV+ω2ψV= 0 ⇒ψV( ) = sin(ω )
ω, ψV(T) = sin(ωT)
ω,(40)
while o he ee pa icle,
¨
ψ0= 0 ⇒ψ0( )= , ψ0(T) = T. (41)
Thus
R≡de ˆ
OV
de ˆ
O0
=sin(ωT)
ωT .(42)
The p opaga o p e ac o is in e sely p opo ional o he squa e oo o he de e minan .
No malizing by he known ee-pa icle p e ac o
K ee(T) = m
2πiT ,(43)
we ob ain
KHO(T) = K ee(T)R−1/2= m
2πiT sωT
sin(ωT)= mω
2πi sin(ωT),(44)
which is p ecisely he Mehle ke nel. This con i ms ha he FM2de e minan limi ep oduces
he exac quan um dynamics o he oscilla o in eal ime.
8
5 Ve i ica ion II: (1+1)D Casimi E ec om Ma ix T ace
We nex show how he FM2ma ix ace yields he s anda d (1 + 1)-dimensional Casimi
ene gy
ECas =−π
24L(45)
wi hou in oking ad-hoc egula iza ion such as ζ- unc ion icks.
5.1 FM2Ma ix Cons uc ion and Ene gy De ini ion
Conside a massless scala ield con ined o an in e al x∈[0, L] wi h Di ichle bounda y
condi ions ϕ(0) = ϕ(L) = 0. The spa ial eigenmodes a e
kn=nπ
L, n = 1,2, . . . (46)
In FM2, he acuum ene gy is ob ained om he imagina y pa o he e ec i e ac ion,
E ac =−1
TIm Γe ∼1
2X
n
ωn (ωn/ΛGabo ),(47)
whe e ωn=knand is a smoo h cu o unc ion de e mined by he Gabo ame, sa is ying
(0) = 1, (ω)→0 apidly as ω→ ∞.(48)
The physical Casimi ene gy is de ined as he di e ence be ween he ene gy in he p esence o
pla es (disc e e spec um) and he e e ence ene gy o ee space (con inuous spec um):
ECas = lim
Λ→∞
1
2"∞
X
n=1
F(n)−Z∞
0
F(n)dn#, F(n)≡ωn (ωn/Λ),(49)
wi h ωn=nπ/L.
5.2 Eule –Maclau in Expansion
The di e ence be ween he sum and he in eg al can be e alua ed using he Eule –Maclau in
o mula: ∞
X
n=0
F(n)−Z∞
0
F(n)dn =1
2F(0) −B2
2! F′(0) + B4
4! F(3)(0) −··· ,(50)
whe e B2= 1/6 is he second Be noulli numbe .
In ou case,
F(n) = nπ
L nπ
LΛ.(51)
Nea n= 0, he cu o sa is ies (0) = 1 and ′(0) = 0, so
F(0) = 0, F′(0) = π
L (0) = π
L.(52)
The highe de i a i es F(k)(0) in ol e de i a i es o and a e supp essed by powe s o 1/Λ;
hey ei he anish as Λ → ∞ o con ibu e only L-independen cons an s ha can be abso bed
in o a bulk eno maliza ion.
Keeping only he uni e sal ini e piece,
∞
X
n=0
F(n)−Z∞
0
F(n)dn ≃ −B2
2F′(0) = −1
12
π
L.(53)
9