Ruprecht-Karls-Universität Heidelberg

Institut für
Angewandte Mathematik

Prof. Dr. Claus Gerhardt

Institut für Angewandte Mathematik
Im Neuenheimer Feld 205
D-69120 Heidelberg

Tel. + 49 (0) 62 21 - 54 14100 (Sekretariat)


Area of Research: Partial Differential Equations, Differential Geometry and General Relativity



A unified quantum theory incorporating the four fundamental forces of nature is one of the major open problems in physics. The Standard Model combines electro-magnetism, the strong force and the weak force, but ignores gravity. The quantization of gravity is therefore a necessary first step to achieve a unified quantum theory.

General relativity is a Lagrangian theory, i.e., the Einstein equations are derived as the Euler-Lagrange equation of the Einstein-Hilbert functional

   (R- 2Λ ),

where N = N n+1  , n ≥ 3  , is a globally hyperbolic Lorentzian manifold, R  the scalar curvature and Λ  a cosmological constant. We also omitted the integration density in the integral. In order to apply a Hamiltonian description of general relativity, one usually defines a time function x0  and considers the foliation of N  given by the slices

M (t) = {x = t}.

We may, without loss of generality, assume that the spacetime metric splits

ds2 = - w2(dx0)2 + gij(x0,x)dxidxj,

cf. [2, Theorem 3.2]. Then, the Einstein equations also split into a tangential part

Gij + Λgij = 0

and a normal part

Gαβνανβ - Λ = 0,

where the naming refers to the given foliation. For the tangential Einstein equations one can define equivalent Hamilton equations due to the groundbreaking paper by Arnowitt, Deser and Misner [1]. The normal Einstein equations can be expressed by the so-called Hamilton condition

H = 0,

where H is the Hamiltonian used in defining the Hamilton equations. In the canonical quantization of gravity the Hamiltonian is transformed to a partial differential operator of hyperbolic type Hˆ and the possible quantum solutions of gravity are supposed to satisfy the so-called Wheeler-DeWitt equation

Hˆu  = 0

in an appropriate setting, i.e., only the Hamilton condition (0.6) has been quantized, or equivalently, the normal Einstein equation, while the tangential Einstein equations have been ignored.

In [2] we solved the equation (0.7) in a fiber bundle E  with base space S0 ,

S0 = {x = 0} ≡ M (0),

and fibers F (x)  , x ∈ S0  ,

F (x) ⊂ T0,2(S ),
        x   0

the elements of which are the positive definite symmetric tensors of order two, the Riemannian metrics in S0  . The hyperbolic operator ˆH is then expressed in the form

Hˆ= - Δ - (R - 2Λ)φ,

where Δ  is the Laplacian of the DeWitt metric given in the fibers, R  the scalar curvature of the metrics gij(x) ∈ F (x)  , and φ  is defined by

 2   detgij
φ  = detρij,

where ρij  is a fixed metric in S0  such that instead of densities we are considering functions. The Wheeler-DeWitt equation could be solved in E  but only as an abstract hyperbolic equation. The solutions could not be split in corresponding spatial and temporal eigenfunctions.

The underlying mathematical reason for the difficulty was the presence of the term R  in the quantized equation, which prevents the application of separation of variables, since the metrics gij  are the spatial variables. In the paper [4] we overcame this difficulty by quantizing the Hamilton equations instead of the Hamilton condition.

As a result we obtained the equation

- Δu = 0

in E  , where the Laplacian is the Laplacian in (0.10). The lower order terms of Hˆ

(R - 2Λ)φ

were eliminated during the quantization process. However, the equation (0.12) is only valid provided n ⁄= 4  , since the resulting equation actually looks like

- (-- 2)Δu = 0.

This restriction seems to be acceptable, since n  is the dimension of the base space S0  which, by general consent, is assumed to be n = 3  . The fibers add additional dimensions to the quantized problem, namely,

dimF =  n(n+-1)-≡ m + 1.

The fiber metric, the DeWitt metric, which is responsible for the Laplacian in (0.12) can be expressed in the form

       16(n - 1)
ds2 = --------- dt2 + φGABd ξAd ξB,

where the coordinate system is

  a     0 A       A
(ξ ) = (ξ ,ξ ) ≡ (t,ξ ).

The (ξA)  , 1 ≤ A ≤ m  , are coordinates for the hypersurface

M  ≡ M (x) = {(gij) : t4 = detgij(x) = 1,∀ x ∈ S0}.

We also assumed that S0 = ℝn  and that the metric ρij  in (0.11) is the Euclidean metric δij  . It is well-known that M  is a symmetric space

M = SL (n,ℝ)∕SO (n) ≡ G∕K.

It is also easily verified that the induced metric of M  in E  is isometric to the Riemannian metric of the coset space G ∕K  .

Now, we were in a position to use separation of variables, namely, we wrote a solution of (0.12) in the form

u = w(t)v(ξA),

where v  is a spatial eigenfunction of the induced Laplacian of M

- ΔM v ≡ - Δv = (|λ|2 + |ρ|2)v

and w  is a temporal eigenfunction satisfying the ODE

      -1      -2
w +mt   ˙w + μ0t w = 0


     16(n - 1)
μ0 = --------(|λ|2 + |ρ|2).

The eigenfunctions of the Laplacian in G ∕K  are well-known and we chose the kernel of the Fourier transform in G ∕K  in order to define the eigenfunctions. This choice also allowed us to use Fourier quantization similar to the Euclidean case such that the eigenfunctions are transformed to Dirac measures and the Laplacian to a multiplication operator in Fourier space.

In [5] we quantized the Einstein-Hilbert functional combined with the functionals of the other fundamental forces of nature, i.e., we looked at the Lagrangian functional

       ∫           ∫
J = α -N1 (R- 2Λ)-    14γabgμρ2gλρ1Faμρ1Fbρ2λ
      ∫ ˜Ω           Ω˜
    -   { 1gμλγabΦaμΦbλ + V (Φ )}
      ∫˜Ω  2
         1 ˜  μ  a     I  ˜---μ-a-----I     ˜  I
    +  ˜Ω{2[ψIEaγ  (D μψ) + ψIE aγ (D μψ) ]+ m ψIψ },

where αN  is a positive coupling constant, ˜         n+1
Ω ⋐ N = N  and N  a globally hyperbolic spacetime with metric gαβ  , 0 ≤ α,β ≤ n  , where the metric splits as in (0.3).

The functional J  consists of the Einstein-Hilbert functional, the Yang-Mills and Higgs functional and a massive Dirac term.

The Yang-Mills field (Aμ)

Aμ = fcAμ

corresponds to the adjoint representation of a compact, semi-simple Lie group G with Lie algebra 𝔤  . The fc  ,

f  = (f a)
 c    cb

are the structural constants of 𝔤  .

We assumed the Higgs field Φ = (Φa)  to have complex valued components.

The spinor field ψ = (ψIA)  has a spinor index A  , 1 ≤ A ≤ n1  , and a colour index I  , 1 ≤ I ≤ n2  . Here, we suppose that the Lie group has a unitary representation R  such that

tc = R (fc)

are antihermitian matrices acting on   n2
ℂ   . The symbol A μψ  is now defined by

A μψ = tcψA cμ.

There are some major difficulties in achieving a quantization of the functional in (0.24). First we were unable to quantize the corresponding Hamilton equations, hence, we quantized the Hamilton condition which has the form

HG  +HY M + HD  + HH = 0,

where the subscripts refer to gravity, Yang-Mills, Dirac and Higgs. On the left-hand side are the Hamilton functions of the respective fields. They depend on the Riemannian metrics gij  , the Yang-Mills connections and the spinor and Higgs fields. We were not able to quantize the non-gravitational Hamiltonians for arbitrary metrics gij  , but we proposed the following model: Choosing the fiber coordinates as in (0.17) the fiber metrics gij  can be written in the form

gij(x) = t4nσij(x),


σij ∈ M,

cf. [3, Equ. (1.4.103)]. We were able to prove that the non-gravitational Hamiltonians could be expressed in the form

       - 2             - 2           - 2
HYM  = t 3 ˜HYM , HD = t  3H ˜D,  HH  = t 3 ˜HH,

where the embellished Hamiltonians depend on σij  , provided n = 3  and provided that the mass term in the Dirac Lagrangian and the Higgs Lagrangian are slightly modified. The embellished Hamiltonians are then standard Hamiltonians without any modifications. The Hamilton constraint then has the form

H = HG + HY M + HH  + HD
          - 2 ˜      ˜    ˜
  = HG + t 3(HY M + HH + HD )
  ≡ HG + t- 23H˜SM = 0,

where the subscript SM  refers to the fields of the Standard Model or to a corresponding subset of fields.

In the quantization process, we quantized HG  for general σij  but ˜
HSM  only for σij = δij  by the usual methods of QFT. Let v  resp. ψ  be the spatial eigendistributions of the respective Hamilton operators, then, the solutions u  of the Wheeler-DeWitt equation are given by u = wvψ  , where w  satisfies an ODE and u  is evaluated at (t,δij)  in the fibers.

The solutions of the Wheeler-DeWitt equation

Hˆu  = 0

can then be achieved by using separation of variables. We proved:

Theorem 0.1. Let n = 3  , v = eλ,b0   and let ψ  be an eigendistribution of H˜SM when σij = δij  such that

- ΔM eλ,b0 = (|λ| + 1)eλ,b0,

H˜SM  ψ = λ1ψ,    λ1 ≥ 0,

and let w  be a solution of the ODE

    ∂    ∂w    32              32        2
t-m∂t (tm ∂t ) + 3-(|λ|2 + 1)t-2w +-3 α -N1λ1t- 3w
              +  --α-N2Λt2w = 0


u = weλ,b0ψ

is a solution of the Wheeler-DeWitt equation

Hu = 0,

where eλ,b0   is evaluated at σij = δij  and where we note that m = 5  .

We referred to eλ,b0   and ψ  as the spatial eigenfunctions and to w  as the temporal eigenfunction.

Remark 0.2. We could also apply the respective Fourier transforms to - ˜Δeλ,b
      0   resp. H˜SM  ψ  and consider

w ˆeλ,bψˆ

as the solution in Fourier space, where ˆψ  would be expressed with the help of the ladder operators.

The temporal eigenfunctions must satisfy an ODE of the form

     - 1      - 2     2- 2       2
w + 5t w˙+ m1t  w + m 2t 3w +m3t  w = 0,


m1 ≥ 3-,  m2 ≥ 0,  m3 ∈ ℝ.

For simplicity we shall only state the result when m3 = 0  which is tantamount to setting Λ = 0  .

Theorem 0.3. Assume m3 = 0  and m2 > 0  , then the solutions of the ODE (0.41) are generated by

  3√ ------ 3    23 -2
J(2  m1 - 4 i,2m2t )t


    √ ------
J(- 32 m1 - 4i, 32m2t23)t-2,

where J(λ,t)  is the Bessel function of the first kind.

Lemma 0.4. The solutions in the theorem above diverge to complex infinity if t  tends to zero and they converge to zero if t  tends to infinity.

For details of the monograph The Quantization of Gravity [3] published by Springer International click here and for a review of the book by Paulo Moniz for the Mathematical Reviews here.


[1]    R. Arnowitt, S. Deser, and C. W. Misner, The dynamics of general relativity, Gravitation: an introduction to current research (Louis Witten, ed.), John Wiley, New York, 1962, pp. 227–265.

[2]    Claus Gerhardt, The quantization of gravity in globally hyperbolic spacetimes, Adv. Theor. Math. Phys. 17 (2013), no. 6, 1357–1391, arXiv:1205.1427, doi:10.4310/ATMP.2013.v17.n6.a5.

[3]    _________ , The Quantization of Gravity, 1st ed., Fundamental Theories of Physics, vol. 194, Springer, Cham, 2018, doi:10.1007/978-3-319-77371-1.

[4]    _________ , The quantization of gravity: Quantization of the Hamilton equations, Universe 7 (2021), no. 4, 91, doi:10.3390/universe7040091.

[5]    _________ , A unified quantization of gravity and other fundamental forces of nature, Universe 8 (2022), no. 8, 404, doi:10.3390/universe8080404.


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