Path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude.
This formulation has proven crucial to the subsequent development of theoretical physics, because manifest Lorentz covariance is easier to achieve than in the operator formalism of canonical quantization. Unlike previous methods, the path integral allows one to easily change coordinates between very different canonical descriptions of the same quantum system. Another advantage is that it is in practice easier to guess the correct form of the Lagrangian of a theory, which naturally enters the path integrals, than the Hamiltonian. Possible downsides of the approach include that unitarity of the S-matrix is obscure in the formulation. The path-integral approach has proven to be equivalent to the other formalisms of quantum mechanics and quantum field theory. Thus, by deriving either approach from the other, problems associated with one or the other approach go away.
The path integral also relates quantum and stochastic processes, and this provided the basis for the grand synthesis of the 1970s, which unified quantum field theory with the statistical field theory of a fluctuating field near a second-order phase transition. The Schrödinger equation is a diffusion equation with an imaginary diffusion constant, and the path integral is an analytic continuation of a method for summing up all possible random walks.
The path integral has impacted a wide array of sciences, including polymer physics, quantum field theory, string theory and cosmology. In physics, it is a foundation for lattice gauge theory and quantum chromodynamics. It has been called the "most powerful formula in physics", with Stephen Wolfram also declaring it to be the "fundamental mathematical construct of modern quantum mechanics and quantum field theory".
The basic idea of the path integral formulation can be traced back to Norbert Wiener, who introduced the Wiener integral for solving problems in diffusion and Brownian motion. This idea was extended to the use of the Lagrangian in quantum mechanics by Paul Dirac, whose 1933 paper gave birth to path integral formulation. The complete method was developed in 1948 by Richard Feynman. Some preliminaries were worked out earlier in his doctoral work under the supervision of John Archibald Wheeler. The original motivation stemmed from the desire to obtain a quantum-mechanical formulation for the Wheeler–Feynman absorber theory using a Lagrangian as a starting point.
Quantum action principle
In quantum mechanics, as in classical mechanics, the Hamiltonian is the generator of time translations. This means that the state at a slightly later time differs from the state at the current time by the result of acting with the Hamiltonian operator. For states with a definite energy, this is a statement of the de Broglie relation between frequency and energy, and the general relation is consistent with that plus the superposition principle.The Hamiltonian in classical mechanics is derived from a Lagrangian, which is a more fundamental quantity in the context of special relativity. The Hamiltonian indicates how to march forward in time, but the time is different in different reference frames. The Lagrangian is a Lorentz scalar, while the Hamiltonian is the time component of a four-vector. So the Hamiltonian is different in different frames, and this type of symmetry is not apparent in the original formulation of quantum mechanics.
The Hamiltonian is a function of the position and momentum at one time, and it determines the position and momentum a little later. The Lagrangian is a function of the position now and the position a little later. The relation between the two is by a Legendre transformation, and the condition that determines the classical equations of motion is that the action has an extremum.
In quantum mechanics, the Legendre transform is hard to interpret, because the motion is not over a definite trajectory. In classical mechanics, with discretization in time, the Legendre transform becomes
and
where the partial derivative with respect to holds fixed. The inverse Legendre transform is
where
and the partial derivative now is with respect to at fixed.
In quantum mechanics, the state is a superposition of different states with different values of, or different values of, and the quantities and can be interpreted as noncommuting operators. The operator is only definite on states that are indefinite with respect to. So consider two states separated in time and act with the operator corresponding to the Lagrangian:
If the multiplications implicit in this formula are reinterpreted as matrix multiplications, the first factor is
and if this is also interpreted as a matrix multiplication, the sum over all states integrates over all, and so it takes the Fourier transform in to change basis to. That is the action on the Hilbert space – change basis to at time .
Next comes
or evolve an infinitesimal time into the future.
Finally, the last factor in this interpretation is
which means change basis back to at a later time.
This is not very different from just ordinary time evolution: the factor contains all the dynamical information – it pushes the state forward in time. The first part and the last part are just Fourier transforms to change to a pure basis from an intermediate basis.
Another way of saying this is that since the Hamiltonian is naturally a function of and, exponentiating this quantity and changing basis from to at each step allows the matrix element of to be expressed as a simple function along each path. This function is the quantum analog of the classical action. This observation is due to Paul Dirac.
Dirac further noted that one could square the time-evolution operator in the representation:
and this gives the time-evolution operator between time and time. In the representation, the quantity summed over the intermediate states corresponds to a matrix element that is not directly observable. In contrast, in the representation, this quantity is interpreted as being associated with a path. Taking a large power of this operator reconstructs the full quantum evolution between two states: the initial state with a fixed value of and the final state with a fixed value of. The resulting expression can be understood as a sum over paths, where each path contributes a phase given by the quantum action..
Classical limit
Crucially, Dirac identified the effect of the classical limit on the quantum form of the action principle:That is, in the limit of action that is large compared to the Planck constant – the classical limit – the path integral is dominated by solutions that are in the neighborhood of stationary points of the action. The classical path arises naturally in the classical limit.
Feynman's interpretation
Dirac's work did not provide a precise prescription to calculate the sum over paths, and he did not show that one could recover the Schrödinger equation or the canonical commutation relations from this rule. This was done by Feynman.Feynman showed that Dirac's quantum action was, for most cases of interest, simply equal to the classical action, appropriately discretized. This means that the classical action is the phase acquired by quantum evolution between two fixed endpoints. He proposed to recover all of quantum mechanics from the following postulates:
- The probability for an event is given by the squared modulus of a complex number called the "probability amplitude".
- The probability amplitude is given by adding together the contributions of all paths in configuration space.
- The contribution of a path is proportional to, where is the action given by the time integral of the Lagrangian along the path.
Feynman showed that this formulation of quantum mechanics is equivalent to the canonical approach to quantum mechanics when the Hamiltonian is at most quadratic in the momentum. An amplitude computed according to Feynman's principles will also obey the Schrödinger equation for the Hamiltonian corresponding to the given action.
The path integral formulation of quantum field theory represents the transition amplitude as a weighted sum of all possible histories of the system from the initial to the final state. A Feynman diagram is a graphical representation of a perturbative contribution to the transition amplitude.
Path integral in quantum mechanics
Time-slicing derivation
One common approach to deriving the path integral formula is to divide the time interval into small pieces. Once this is done, the Trotter product formula tells us that the noncommutativity of the kinetic and potential energy operators can be ignored.For a particle in a smooth potential, the path integral is approximated by zigzag paths, which in one dimension is a product of ordinary integrals. For the motion of the particle from position at time to at time, the time sequence
can be divided up into smaller segments, where, of fixed duration
This process is called time-slicing.
An approximation for the path integral can be computed as proportional to
where is the Lagrangian of the one-dimensional system with position variable and velocity considered, and corresponds to the position at the th time step, if the time integral is approximated by a sum of terms.
In the limit, this becomes a functional integral, which, apart from a nonessential factor, is directly the product of the probability amplitudes to find the quantum mechanical particle at in the initial state and at in the final state.
Actually is the classical Lagrangian of the one-dimensional system considered,
and the abovementioned "zigzagging" corresponds to the appearance of the terms
in the Riemann sum approximating the time integral, which are finally integrated over to with the integration measure, is an arbitrary value of the interval corresponding to, e.g. its center,.
Thus, in contrast to classical mechanics, not only does the stationary path contribute, but actually all virtual paths between the initial and the final point also contribute.