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The Problem of Time in the Wheeler-DeWitt Equation, and Dark Energy

In quantizing gravity, we have the trouble of missing time. This trouble may most easily be explained in a simplified Wheeler-DeWitt equation, which will be shown in the former half of this article. In the latter half, I will present my tentative idea on this issue. A prime symbol will be used to mean a time derivative because of web limitation.
(Japanese version of this post is available.)

The most popular way to quantize is using canonical formalism. ADM formalism [1] is usually used in handling the canonical formalism of gravity. In ADM formalism, the metric of the spacetime will be written as

  gμν =
   ( -N2+NiNj hij Nj
      Ni   hij )
(A matrix will be denoted in the way above because of web limitation.) Wheeler-DeWitt equation of isotropic space will be considered here, because the problem of time is not different even with the full theory that includes non-isotropic fluctuations. The metric of the isotropic space is
  ds2 = - dt2 + a2(t) ( dr2/(1 - kr2) + r22 + r2 sin2 θ dφ2 ) ,
where k stands for the normalized curvature. This metric is called Friedmann–Lemaître–Robertson–Walker (FLRW) metric, or shortly, Robertson–Walker metric. In ADM formalism, this metric corresponds to
  hij =
   ( a2/(1 - kr2)  0  0
      0     a2r2 0
      0    0  a2r2sin2 θ )       (1)
with Ni = 0. The Lagrangian for this metric is
 L = ∫d3x √{-g} R = ∫d3x N√h ( 3R - K2 + (Kij)2 )
after an tedious calculation, where Kij = hij' /2N で Kij = hikhjlKkl , K = hijKij. The integration gives
  L = 6 (N k a - a a'2 /N) .
(The factor 6 will be abbreviated afterwards.)

To quantize, we usually consider that we need to move to canonical formalism, which is an idea originating in Dirac. The canonical momenta for this system are
  Π = δL/δN' = 0
  π = δL/δa' = - 2 a a' /N .
Here the canonical variable Π is not free and permanently under the condition Π = 0. Such condition is called ``constraint'', and a system with at least one constraint is called ``constrained system''.

How should the constrained systems be treated in canonical formalism? The Dirac's idea is as follows[2]. We begin with defining the Hamiltonian with H = Π N' + π a' - L as usual. Then
  H = N ( k a - π2/a ) .
Because the condition Π = 0 should hold after time evolution under this Hamiltonian, we have another constraint condition:
  0 = Π' = dH/dN = ( k a - π2/a ) ≡ φ .
Such conditions that come from the consistency of the constraints over the time are called ``secondary constraints''. Repeating such a procedure until we have no more secondary constraints, we will have a set of equations that is equivalent to Euler-Lagrange equations. We are happy with this result in classical mechanics.

However, we have trouble in quantum mechanics. Wheeler and DeWitt considered that the constraints should hold on any quantum state |ψ〉[3]:
  Π |ψ〉= 0 ,
  φ |ψ〉= 0 .
In the case of quantum gravity, we have
  H |ψ〉= 0
because of those condition. No time evolution will occur under those conditions, which effectively mean that there is no time. This is the problem of time in quantum gravity.

Here I start presenting my tentative idea for the standard description of the problem above. I suggest introducing a variable T = ∫dt N instead of g00 = N to solve the problem of time. This variable is subjective time for the observer in the system. With this variable, the Lagrangian is
  L = (dT/dt)( - a (dt/dT)2 a'2 + k a ) .
This way of parametrization is not strange since this is the same as that of string theory. (What would occur to string quantization if we impose constraints on the states instead of the customary method of gauge fixing? This question is answered in [4], which have less difficulty than the customary method.) This variable T may be regarded as a local coordinate of the spacetime, and also regarded as a chart of the spacetime manifold.

Let us move to canonical formalism.
  Π = δL /δT' = a (dt/dT)2 a'2 + k a ,
  π = δL/δa'= - 2 a a' /N .
Since we have condition
  Π + π2/4a = ka
irrespective of time, this system is a constrained system. With the definition H = Π N' + π a - L, we have
  H = 0 .
It can be easily seen that we have no more constraints. Using the WdW idea, we have an equation for the wave function:
  (∂/∂T + π2/4a - ka) |ψ〉= 0 .
This equation may be physically interpreted as a Schrödinger equation with its time variable T. (The variable T gives physically meaningful equation in contrast to taking t and N as variables, which have gauge indeterminancy.) Why taking variable T was better? The answer may be that gravity is rather a dynamics of manifolds (or local coordinates) than a dynamics of the metric. If we consider that our curved space owes to embedding into higher-dimensional space, this choice of variable is more natural.

Let us see the physical outcomes of this new method. The equation of motion out of this method is
  a a'2 / N2 + ka = λa3/3 + Π .,
On the other hand, the customary method gives
  aa'2 / N2 + ka = λa3/3
The solution of the ordinary method a = exp √{λ/3} t , for the case of k=0 for brevity, is known as de Sitter space is solution of
  a'2/a2 = λ/3 .
In contrast, the differential equation of our method is
  a'2/a2 = λ/3 + Π/a ,
and we easily see that the universe tend to expand more than the conventional theory.

Thus our method gives a little bit different outcomes from that of Einstein equations. Usually one may want to abandon such a theory. Though we have unusual situation where we are considering on the ``dark energy'' to explain excessive tendency of expansion, which can include some deviation from Einstein equations. Our method has a virtue of explaining that fact with minimal change without adding any new matter nor any new mechanism.

[1] Arnowitt, R.; Deser, S.; Misner, C. (1959). "Dynamical Structure and Definition of Energy in General Relativity". Physical Review 116 (5): 1322?1330.

[2'] P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva Univ., 1964)

[3] Usually, we handle quantization of the constrained systems with gauge fixing in Lagrangian, or gauge fixing in canonical formalism to bring it to the Dirac brackets. The Wheeler-DeWitt idea of using the constraint equations as if they are Schrödinger equations is only seen in the context of quantum gravity, as far as the author knows.

[4] T. Mogami, Quantization of Nambu-Goto Action in Four Dimensions, arXiv:1005.2726.

P.S. My question about this tentative idea is if it is generally covariant since what is lost is R00 component of the Einstein equations sparing other components like Rii. However, since it is based on the invariant action of dynamical variables which is a local coordinate of a chart of the space-time manifold, it cannot break general covariance. I would like to investigate this point sometime.

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重力を量子化するさいは時間の問題が起こります。以下ではこの問題をWheeler-DeWitt方程式で端的に説明します。そして後半ではこの問題についての私の試行的アイデアについて説明します。だたし以下では普通と違って' primeが時間微分を表している事に注意してください。

  gμν =
   ( -N2+NiNj hij Nj
      Ni   hij )
と書きます。(行列は以上のように表記する事にします。)ここでは一様等方な宇宙のWheeler-DeWitt方程式を考えます。一様等方以外のfluctuationも考えたfullの理論でも、時間の問題は変わらないからです。その計量は規格化された局率を k として
  ds2 = - dt2 + a2(t) ( dr2/(1 - kr2) + r22 + r2 sin2 θ dφ2 )
です。この計量は「フリードマン・ルメートル・ロバートソン・ウォーカー計量」または省略して、FLRW計量ないし「ロバートソン・ウォーカー計量」と呼ばれている。この計量はADM形式の Ni = 0,
  hij =
   ( a2/(1 - kr2)  0   0
      0    a2r2   0
       0   0  a2r2sin2 θ )       (1)
  L = ∫d3x √{-g} R = ∫d3x N√h ( 3R - K2 + (Kij)2 )
となります。ここでは Kij = hij' /2N で Kij = hikhjlKkl , K = hijKij です。さらに積分を実行すると
  L = 6 (N k a - a a'2 /N)

  Π = δL/δN' = 0
  π = δL/δa' = - 2 a a' /N
となります。ここでΠは自由でなく、Π = 0 という条件が恒常的に成り立ちます。こののような条件を拘束条件、そしてこのように正準変数のうちに自由ではないようなものがある系を拘束系といいます。

拘束系の正準理論をどのようにしたら良いのか、Diracの考え[2]を今の場合にそって簡略に紹介します。まずHamiltonianは普通に H = Π N' + π a' - L によって定義します。すると
  H = N ( k a - π2/a )
このHamilton下でΠ = 0が矛盾無くずっと成り立っていなければなりませんから、
  0 = Π' = dH/dN = ( k a - π2/a ) ≡ φ

しかし量子論では問題が起きます。Wheeler-DeWitt方程式では任意の状態 |ψ〉に対して拘束条件が成り立つと考えます[3]:
  Π |ψ〉= 0
  φ |ψ〉= 0
  H |ψ〉= 0

以上はこの問題の標準的な理解ですが、ここから試論にはいります。この時間の問題に対して、ここでは g00 = N にかわり ∫dt N = T という変数を導入することをを提案します。これは、系内の存在に取っての主観時間のようなものです。この変数によりLagrangianは
  L = (dT/dt)( - a (dt/dT)2 a'2 + k a )
となります。これは1次元粒子や2次元物体であるstring theoryでのパラメトライズの仕方と同じで奇矯な物ではありません。(普通にゲージ固定するのでなく、拘束条件を状態に課すというアイデアをstringに適用したものは[4]となります。つまり普通のstringのパラメトライズの方法と普通のWdWのアイデアをつないだ物は普通のstring theoryより困難が少なくなります。)この変数Tは時空の局所座標とみなすこともでしますし、時空多様体のchartとみなすこともできます。

  Π = δL /δT' = a (dt/dT)2 a'2 + k a
  π = δL/δa'= - 2 a a' /N
  Π + π2/4a = ka
という条件が時間によらず成り立つので、この系は拘束系です。さらに定義 H = Π N' + π a - L により
  H = 0
です。これ以上のconsistency conditionがでることはないことは簡単にわかります。ここでWdWのアイデアを適用すると波動関数に関する方程式は
  (∂/∂T + π2/4a - ka) |ψ〉= 0
となります。見ての通り、時間変数TについてのSchroedinger方程式という物理的解釈ができます。ゲージ変換による不定性をもつ t とNとはちがい、Tについては物理的に意味のある式になる訳です。

  a a'2 / N2 + ka = λa3/3

  a a'2 / N2 + ka = λa3/3 + Π

となることです。(簡単のためにk=0のときのみ書きました。)前者の解はde Sitter spaceという名で知られていて
  a'2/a2 = λ/3 
の解 a = exp √{λ/3} t が答えなのに対して、後者は
  a'2/a2 = λ/3 + Π/a



[1] Arnowitt, R.; Deser, S.; Misner, C. (1959). "Dynamical Structure and Definition of Energy in General Relativity". Physical Review 116 (5): 1322?1330.

[2'] P.A.M. Dirac, Lectures on Quantum Mechanics (Belfer Graduate School of Science, Yeshiva Univ., 1964)

[3] 通常、拘束系の量子化は、Lagrangianでゲージ固定をしたり、正準形式でゲージ固定してDirac括弧に持っていったりします。拘束条件をSchroedinger方程式のように考えるというWheeler-DeWittのアイデアは、量子重力以外の文脈では見かけた事がありません。

[4] T. Mogami, Quantization of Nambu-Goto Action in Four Dimensions, arXiv:1005.2726.

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