Wheeler-DeWitt equationequation
In quantum gravity, time disappears as a fundamental quantity. The Wheeler–DeWitt equation is a direct result of applying the laws of quantum mechanics to Einstein's theory of gravitation. John Wheeler and Bryce DeWitt formalized quantum theory in the so-called canonical approach in 1967. It turned out that the wave function of the universe (the amplitude of all possible geometries) is independent of time — it is a static eigenvalue equation where the Hamiltonian operator yields zero. That is, at the deepest level, there is no evolution: all moments of time 'are present' simultaneously. Analogy: a vast library where, instead of books, three-dimensional slices of space are kept, and the equation is the rule binding them into a single four-dimensional framework.
How it works
Arises from attempts to unify quantum mechanics (the behavior of particles at the micro level) and general relativity (the geometry of space-time). Applied in models of the early universe, where space-time was quantum. In practice, it helps theorists calculate the probabilities of various scenarios for the birth of the universe, for example, from quantum tunneling.
💡 Because time is absent from the equation, some physicists jokingly call it 'the equation where nothing happens.'
\left( \mathcal{G}_{ijkl} \frac{\delta^2}{\delta h_{ij} \delta h_{kl}} + \sqrt{h} \, ^{(3)}R \right) \Psi[h_{ij}] = 0
𝒢_ijkl — суперметрика (метрика на пространстве трёхмерных метрик), h_ij — трёхмерная метрика на пространственноподобном срезе, h = det(h_ij) — её определитель, ⁽³⁾R — скалярная кривизна трёхмерного среза, Ψ[h_ij] — волновая функция Вселенной (функционал, задающий амплитуду вероятности обнаружить пространство с заданной геометрией). Оператор 𝒢_ijkl·δ²/δh_ij·δh_kl — кинетический член, а √h·⁽³⁾R — потенциальный член. Нулевой результат отражает отсутствие внешней временной координаты.