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Hadamard renormalisation for charged scalar fields
Visakan Balakumar
Supervised by Elizabeth Winstanley
School of Mathematics and StatisticsUniversity of Sheffield
CRAG seminar on 3rd June 2020
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Introduction
The stress-energy tensor of a QFT gives important information aboutthe particle content, or flux of energy.
We would like to explicitly calculate 〈Tµν〉 to use as a source term inEinstein’s semiclassical field equations.
Gµν := Rµν −1
2gµνR = 〈Tµν〉 . (1)
〈Tµν〉 is formed from products of field operators evaluated at thesame spacetime point x , causing it to be formally divergent.
Need to renormalise 〈Tµν〉 - we do this by Hadamard renormalisationdeveloped in conjunction with Wald’s axioms.
Uses point-splitting approach that involves taking one field operatorto a small, but finite, distance away.
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Geometrical preliminaries
Hadamard parametrix depends upon Synge’s world function σ(x , x ′).
Defined as half the square of the geodesic distance between x and x ′:
2σ = σ ;µσ ;µ , (2)
where σ ;µ = ∇µσ.
Also require van Vleck-Morette determinant ∆(x , x ′), which gives therate at which geodesics converge or diverge from each other.
In D dimensions, it is related to the world function by:
∇µ∇µσ = D − 2∆−12 ∆
12;µσ
;µ, (3)
with boundary condition limx→x ′ ∆(x , x ′) = 1.
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Hadamard expansion of Feynmann propagator
Choosing an appropriate vacuum state |ψ〉, 〈Tµν〉 is defined to be:
〈ψ|Tµν(x) |ψ〉 = limx ′→x
Tµν(x , x ′)[− i GF (x , x ′)
], (4)
where GF is the Feynmann Green’s function.
In D = 4, short-distance singularity structure of GF given by:
GF(x , x ′) =i
8π2
[U(x , x ′)
[σ + iε]+ V (x , x ′) ln[σ + iε] + W (x , x ′)
], (5)
where U(x , x ′), V (x , x ′) and W (x , x ′) are biscalar functions, regular asx ′ → x , that admit power series expansions in σ.
Explicitly we have U = U0 and V = Σ∞n=0Vnσn.
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Hadamard renormalisation
U(x , x ′) and V (x , x ′) are uniquely-determined geometric quantities.
They contain the singular behaviour, so we can write GF as:
GF(x , x ′) = GFsing(x , x ′) + GF
reg(x , x ′) . (6)
Then 〈Tµν〉ren is given by:
〈ψ|Tµν(x) |ψ〉ren = limx ′→x
Tµν[− i(GF(x , x ′)− GF
sing(x , x ′))]. (7)
Means we need to find U0(x , x ′) and Vn(x , x ′) explicitly.
Decanini and Folacci have given the general Hadamard procedure fora neutral scalar field in 2008.
We would like to extend this for a scalar field with arbitrary charge q.
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Governing equations for Hadamard coefficients
We employ a covariant Taylor expansion method. With a fixed, classicalbackground gauge field Aµ, the gauge covariant derivative is defined as:
Dµ = ∇µ − iqAµ . (8)
Then, the inhomogenous Klein-Gordon equation is given by:
(DµDµ −m2 − ξR)GF(x , x ′) = − 1√
−g(x)δ4(x − x ′) , (9)
where g(x) = det[ gµν(x) ].
Substituting Hadamard form of GF into (9) gives us expression interms of σ.
Equating powers of σ gives us equations U and V must satisfy.
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Finding U(x , x ′)
We require U0 to fourth order to obtain the RSET.
Collecting terms proportional to σ−2 gives equation for U0:
U0 ;µ σ;µ −
((∆−
12 )∆
12 ;µσ
;µ)U0 − iqAµU0 σ
;µ = 0 . (10)
In the uncharged case, the exact solution to (10) is U0 = ∆12 .
To find U0 in the charged case, we expand as a covariant Taylorexpansion:
U0 = U00 + U01µσ;µ + U02(µν)σ
;µσ;ν + U03(µνρ)σ;µσ;νσ;ρ
+ U04(µνρτ)σ;µσ;νσ;ρσ;τ + . . .
(11)
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Coefficients of U0(x , x ′) I
Considering powers of σ in the equation for U0, we find the followingexpressions for the coefficients:
U00 = 1; (12)
U01µ = iqAµ; (13)
U02(µν) =1
12Rµν −
1
2iq∇(µAν) −
1
2q2A(µAν); (14)
U03(µνρ) = − 1
24R(µν;ρ) +
1
12iq A(µRνρ) +
1
6iq∇(µ∇νAρ)
+1
2q2A(µ∇νAρ) −
1
6iq3A(µAνAρ). (15)
The expression for U04(µνρτ) is too long to fit on the slide!
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Coefficients of U0(x , x ′) II
U04µνρτ =1
80∇(µ∇νRρτ) +
1
360Rθ
(µ|φ|νRφρ|θ|τ) +
1
288R(µνRρτ)
− 1
24iqA(µ∇νRρτ) −
1
24iq(∇(µAν
)Rρτ) −
1
24q2A(µAνRρτ)
− 1
24iq∇(µ∇ν∇ρAτ) −
1
6q2A(µ∇ν∇ρAτ)
− 1
8q2(∇(µAν
) (∇ρAτ)
)+
1
4iq3A(µAν∇ρAτ)
+1
24q4AµAνAρAτ . (16)
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Coefficients of U0(x , x ′) III
We can simplify the expressions by writing them in terms of the gaugecovariant derivative Dµ:
(DµU0)σ ;µ −((∆−
12 )∆
12 ;µσ
;µ)U0 = 0 . (17)
U00 = 1; (18)
U01µ = iqAµ; (19)
U02(µν) =1
12Rµν −
iq
2D(µAν); (20)
U03(µνρ) = − 1
24R(µν;ρ) +
iq
6D(µDνAρ) +
iq
12A(µRνρ); (21)
U04(µνρτ) =1
80R(µν;ρτ) +
1
360Rθ
(µ|φ|νRφρ|θ|τ) +
1
288R(µνRρτ)
− iq
24D(µDνDρAτ) −
iq
24D(µ[AνRρτ)]. (22)
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Finding V (x , x ′)
Collecting terms proportional to lnσ gives recursion relation for Vn:
2(n + 1)2Vn+1 + 2(n + 1) (DµVn+1)σ ;µ − 2(n + 1)Vn+1∆−12 ∆
12
;µσ;µ
+ (DµDµ −m2 − ξR)Vn = 0 .(23)
We require V0 to second order to obtain the RSET:
V0 = V00 + V01µσ;µ + V02(µν)σ
;µσ;ν + . . . (24)
We only require lowest order term of V1 to find 〈Tµν〉ren:
V1 = V10 + . . . (25)
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Coefficients of V0
Then for the coefficients of V0 we obtain:
V00 =1
2
[m2 +
(ξ − 1
6
)R
]; (26)
V01µ = −1
4
(ξ − 1
6
)R;µ +
iq
2
[m2 +
(ξ − 1
6
)R
]Aµ −
iq
12∇αFαµ;(27)
V02(µν) =1
24
[m2 +
(ξ − 1
6
)R
]Rµν +
1
12
(ξ − 3
20
)R;µν −
1
240�Rµν
+1
180Rα
µRαν −1
360RαβRαµβν −
1
360Rαβγ
µRαβγν
− iq
4
[m2 +
(ξ − 1
6
)R
]D(µAν) −
iq
4
(ξ − 1
6
)A(µR;ν)
− q2
24Fα
µFνα −q2
12A(µ∇αFν)α −
iq
24∇(µ∇αFν)α. (28)
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Coefficient of V1
For the lowest order term in V1 we obtain:
V10 =1
8
[m2 +
(ξ − 1
6
)R
]2
− 1
24
(ξ − 1
5
)�R − 1
720RαβRαβ
+1
720RαβγδRαβγδ +
q2
48FαβFαβ. (29)
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Real symmetric biscalars in the uncharged case
Given a real, symmetric biscalar K (x , x ′) = K (x ′, x), with a covariantTaylor expansion given by:
K (x , x ′) = k0(x) + k1µ(x)σ ;µ + k2(µν)(x)σ ;µσ ;ν + k3(µνρ)σ;µσ ;νσ ;ρ + . . .
(30)we can express odd coefficients in terms of even ones. At lowest orders:
k1µ = −1
2k0;µ , (31)
k3(µνρ) = −1
2k2(µν;ρ) +
1
24k;(µνρ) . (32)
In the case of U0, we have U00 = 1 while U01µ = 0. Also:
U03(µν) = − 1
24R(µν;ρ) = −1
2∇(ρ
(1
12Rµν)
)= −1
2U02(µν;ρ) +
1
24U0;(µνρ) .
(33)
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Complex sequisymmetric biscalars in the charged case
In the charged case, U0 and Vn are complex biscalars sequisymmetricin the exchange of x and x ′, which satisfy:
K (x , x ′) = K ∗(x ′, x) . (34)
The expressions relating even and odd coefficients become:
<[k1µ
]= −1
2k0;µ , (35)
<[k3(µνρ)
]= −1
2R[k2(µν;ρ)
]+
1
24k;(µνρ) . (36)
Considering the charged corrections to U0, we can verify that:
<[U03(µνρ)
]=
q2
2A(µ∇νAρ) = −1
2∇(ρ
(− q2
2AµAν)
)= −1
2<[U02(µν;ρ)
].
(37)
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Renormalised expectation values
We subtract the divergent parts from GF(x , x ′) to give a regularisedGreen’s function:
−i GFreg(x , x ′) = −i
[GF(x , x ′)− GF
sing(x , x ′)]
= αW (x , x ′) . (38)
W (x , x ′) depends on the quantum state under consideration. We canexpand it as:
W (x , x ′) = w0(x) + w1µσ;µ + w2µνσ
;µσ ;ν + w3µνρσ;µσ ;νσ ;ρ + . . . (39)
The vacuum polarisation is then given as:
〈ΦΦ†〉ren = limx ′→x
R[−i GF
reg(x , x ′)]
= αw0(x) . (40)
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Identities concerning W (x , x ′) I
W (x , x ′) satisfies the equation:
0 = [DµDµ − (m2 + ξR)]W + 2[σ ;µDµ + 3]V1 +O(σ) . (41)
We can generate equations that W must satisfy in terms of V byinserting their expansions. At zeroth order in σ we have:
0 = DµDµw0 + 2Dµw1µ + 2gµνw2µν −
(m2 + ξR
)w0
+ 2 (p + 1)V10 . (42)
Taking real and imaginary parts of (42), we obtain:
0 = 2gµν<[w2µν ] + 2qAµ=(w1µ)− [m2 + ξR + q2AµAµ]w0 (43)
+ 2 (p + 1)V10 ,
0 = ∇µ= (w1µ)− qAµ∇µw0 − q (∇µAµ)w0 . (44)
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Identities concerning W (x , x ′) II
Evaluating (41) to order σ12 , we obtain:
0 = DµDµw1α + 4Dµw2αµ + 6gµνw3αµν +
1
3Rµ
αw1µ
−(m2 + ξR
)w1α. (45)
We only require the real part of (45):
0 = 2∇µ< (w2αµ) + qAµ∇µ= (w1α) + 2q(∇(αA
µ)=(w1µ)
)− 1
4∇α�w0 −
1
2Rµ
αw0;µ −[
1
2ξR;α + q2Aµ∇αAµ
]w0
+ ∇αV10 . (46)
We can now generate expressions for 〈Jµ〉ren and 〈Tµν〉ren in terms ofthe coefficients of W (x , x ′).
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Renormalised expectation value of the current
The classical current Jµ for a charged complex scalar field Φ is:
Jµ =iq
8π[Φ∗DµΦ− Φ (DµΦ)∗] = − q
4π= [Φ∗DµΦ] . (47)
Therefore the renormalised expectation value is given by:
〈Jµ〉ren = − q
4πlimx ′→x
={Dµ[−iGR(x , x ′)
]}=αq
4π{qAµw0 −= [w1µ]} .
(48)
We require (48) to be conserved, that is:
0 = ∇µ〈Jµ〉ren =q
4π{q (∇µAµ)w0 + qAµ∇µw0 −∇µ= [w1µ]} , (49)
which holds from (43).
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Renormalised stress-energy tensor I
The RSET is given by:
〈Tµν〉 = limx ′→x
<{Tµν(x , x ′)
[−iGR(x , x ′)
]}, (50)
where Tµν is a second order differential operator given by:
Tµν = (1− 2ξ) gνν′DµD
∗ν′ +
(2ξ − 1
2
)gµνg
ρτ ′DρD∗τ ′ − 2ξDµDν
+ 2ξgµνDρDρ + ξ
(Rµν −
1
2gµνR
)− 1
2m2gµν . (51)
It is unique only up to the addition of a local conserved tensor:
〈Tµν〉ren = α limx ′→x
<[Tµν(x , x ′)W (x , x ′)
]+ Θµν , (52)
where Θµν will be constrained by considering the divergence of 〈Tµν〉ren.
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Renormalised stress-energy tensor II
Evaluating (52), we obtain:
〈Tµν〉ren = α
{−2< (w2µν)− 2qA(µ=
(w1ν)
)−(ξ − 1
2
)w0;µν
+(ξRµν + q2AµAν
)w0 + gµν [gρτ< (w2ρτ ) + qAρ= (w1ρ)
+
(ξ − 1
4
)�w0 −
1
2
(m2 + ξR + q2AρA
ρ)w0
]}+ Θµν . (53)
This expression is manifestly symmetric in µ and ν and reduces to theuncharged case when Aµ = 0.
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Conservation of the RSET I
Taking the divergence of (53) and using (43) to (45), we get:
∇µ〈Tµν〉ren = −2α∇νV10 + 4πFµν〈Jµ〉ren +∇µΘµν . (54)
We can define:
Θµν = 2αgµνV10 + Θµν . (55)
where Θµν is a local conserved tensor, giving the expected renormalisationambiguity in the RSET.
However, we now have:
∇µ〈Tµν〉ren = 4πFµν〈Jµ〉ren (56)
leading to the nonconservation of the RSET of the charged scalar field.
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Conservation of the RSET II
There are two matter fields in our system - the quantum chargedscalar field and a classical background electromagnetic field.
Only the total stress-energy tensor will be conserved.
The stress-energy tensor due to the electromagnetic field is:
TFµν = FµρFν
ρ − 1
4gµνFρτF
ρτ . (57)
Taking the divergence gives:
∇µTFµν = Fνρ∇µF
µρ = 4πFνρ〈Jρ〉ren, (58)
where we have used Maxwell’s equation 0 = ∇[µFρτ ] (and the secondequality follows from the semiclassical Maxwell equation).
Since the electromagnetic field Fµν is antisymmetric, the totalstress-energy tensor TF
µν + 〈Tµν〉ren is conserved, as required.
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Renormalisation ambiguities I
There is an additional ambiguity due to the choice of renormalisationlength scale ` in the Hadamard parametrix.
This leads to an ambiguity in the Hadamard coefficient W (x , x ′)corresponding to the freedom to make the replacement:
W (x , x ′)→W (x , x ′) + V (x , x ′) ln `2. (59)
Ambiguities in 〈ΦΦ†〉ren, 〈Jµ〉ren and 〈Tµν〉ren therefore arise.
For the scalar condensate, we find:
〈ΦΦ†〉ren → αw0 + V00 ln `2. (60)
From (26), this depends on R, m and ε but not the electromagneticpotential.
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Renormalisation ambiguities II
For the current, we find:
〈Jµ〉ren →αq
4π
{qAµw0 −= [w1µ] +
q
12(∇ρFρµ) ln `2
}. (61)
The current acts as a source for the semiclassical Maxwell equationsthrough ∇µF
µν = 4π 〈Jν〉ren.
This corresponds to the constant renormalisation of the permeabilityof free space µ0.
For the stress-energy tensor, we find:
〈Tµν〉ren → 〈Tµν〉ren + Ψµν ln `2, (62)
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Renormalisation ambiguities III
Ψµν = α
{1
2
(ξ − 1
6
)[m2 +
(ξ − 1
6
)R
]Rµν +
1
120�Rµν
− 1
2
(ξ2 − 1
3ξ +
1
30
)R;µν −
1
90Rα
µRαν +1
180RαβRαµβν
+1
180Rαβγ
µRαβγν +q2
12Fα
µFνα + gµν
{1
720RαβRαβ
− 1
720RαβγδRαβγδ +
1
2
(ξ2 − 1
3ξ +
1
40
)�R
− 1
8
[m2 +
(ξ − 1
6
)R
]2
+q2
48FαβFαβ
}}. (63)
Curvature terms correspond to higher-order corrections to thegravitational action giving rise to Einstein’s semiclassical equations.Gauge field corrections ∼ to TF
µν so it corresponds to renormalisationof the gravitational constant G .
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Trace anomaly
The general expression for the trace anomaly is given by:
〈Tµµ 〉ren = −α
{m2w0 − 3 (ξ − ξc)�w0 − 2V10
}+ gµνΨµν . (64)
This simplifies to:
〈Tµµ 〉ren =
1
4π2
[1
720�R − 1
720RαβRαβ +
1
720RαβγδRαβγδ −
q2
48FαβFαβ
].
(65)
The gauge correction to the trace anomaly depends only on Fµν andalso arises in Minkowski spacetime.
This correction agrees with DeWitt-Schwinger method as welladiabatic regularisation on cosmological spacetimes.
Non-trivial check on the validity of our results and provides strongevidence for the equivalence of these approaches.
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Summary and outlook
Generalised the Hadamard procedure for charged scalar fields andexplicitly calculated Hadamard coefficients in D = 4.
Found that the trace anomaly of the RSET is modified by a termproportional to the electromagnetic field strength.
We wish to look at some specific examples, such as a charged scalarfield propagating in Reissner-Nordstrom spacetime.
Equivalence of Hadamard and adiabatic approaches have been provenfor a neutral scalar field - extension to the charged case.
The full work can be accessed via:https://arxiv.org/pdf/1910.03666.pdf
Thanks for listening!Visakan Balakumar (Sheffield) Hadamard renormalisation for charged scalar CRAG 28 / 28