- #1
jostpuur
- 2,116
- 19
Problem:
How do you quantize a two dimensional system defined by the Lagrange's function
[tex]
L=\dot{x}y - x\dot{y} - x^2 - y^2 ?
[/tex]
This is a non-trivial task, because the system has some pathology. Classically the equations of motion are
[tex]
\dot{x}(t) = y(t)
[/tex]
[tex]
\dot{y}(t) = -x(t),
[/tex]
and for arbitrary initial configuration x(0), y(0), the solution is
[tex]
\left[\begin{array}{c}
x(t) \\ y(t) \\
\end{array}\right]
= \left[\begin{array}{rr}
\cos t & \sin t \\
-\sin t & \cos t \\
\end{array}\right]
\left[\begin{array}{c}
x(0) \\ y(0) \\
\end{array}\right]
[/tex]
Alternatively, the L, EOM, and solution can be written more compactly with complex numbers:
[tex]
L=\textrm{Im}(\dot{z}^* z) - |z|^2
[/tex]
[tex]
\dot{z} = -iz
[/tex]
[tex]
z(t)=e^{-it}z(0)
[/tex]
Usually the equations of motion define the second time derivatives of the coordinates, so that both coordinates and velocities are needed for unique solution. With this system velocities are not independent of the coordinates, and as consequence the usual quantization procedure doesn't really work.
An attempt with the Schrödinger's equation and Hamiltonian:
The Hamiltonian can be solved to be
[tex]
H=\frac{\partial L}{\partial\dot{x}}\dot{x} + \frac{\partial L}{\partial\dot{y}}\dot{y} - L = x^2 + y^2 = |z|^2
[/tex]
This looks pretty strange Hamiltonian, but it is in fact one of the most obviously conserving quantities in the system, so it could be considered some kind of energy. The SE is then
[tex]
i\partial_t\Psi(t,z) = |z|^2\Psi(t,z),
[/tex]
and solutions are
[tex]
\Psi(t,z) = e^{-i|z|^2 t}f(z).
[/tex]
Clearly something is wrong with this, because the quantum mechanical solutions do not give the classical behavior on the classical limit. The problem is that the canonical momenta are
[tex]
p_x = \frac{\partial L}{\partial\dot{x}} = y
\quad\quad\quad
p_y = \frac{\partial L}{\partial\dot{y}} = -x
[/tex]
so when we should leave parameters x and y untouched, and substitute [itex]p_x\to -i\partial_x[/itex] and [itex]p_y\to -i\partial_y[/itex], the coordinates and momenta get confused. I don't really know where those derivative operators should be put.
An attempt with path integrals and action:
Suppose the system goes from z' to z, in time [itex]t_1-t_0[/itex]. We can parametrize a path
[tex]
z(t) = \frac{(t_1-t)z' + (t-t_0)z}{t_1-t_0},
[/tex]
and compute the action
[tex]
S = \int\limits_{t_0}^{t_1} L(z(t))dt = xy' - yx' - \frac{1}{3}(t_1-t_0)\big( x^2 + xx' + (x')^2 + y^2 + yy' + (y')^2\big),
[/tex]
but this cannot be used to define time evolution with
[tex]
\Psi(t+\Delta t, z) = N \int dx'\;dy'\; e^{iS}\Psi(t,z')\;+\; O(\Delta t^2)
[/tex]
as usual, because there is no
[tex]
\propto \quad\frac{1}{t_1-t_0},\quad\quad\textrm{or}\quad \frac{1}{(t_1-t_0)^2}
[/tex]
kind of terms in the S as usual. Such terms are necessary to make actions for large spatial transition in small time to become infinite, and to produce necessary oscillation in the path integral.
At this point I'm out of ideas.
Motivation
This is not an unphysical example. Even though physical systems usually have EOM containing second order time derivatives, this is not always the case, and the classical Dirac field is the most obvious counter example, since Dirac equation contains only first order derivatives. Also, any initial [itex]\psi(0,x)[/itex] alone always fixes the time evolution uniquely.
One question that stroke me already in the beginning of the studies of QFT was, that if the quantization of the Klein-Gordon field is based on the quantization of harmonic oscillators, then what is the quantization of the Dirac's field based on? This question is easily answered by writing the Lagrangian
[tex]
L = \int d^3x\; \textrm{Re}\Big(i\overline{\psi}(x)\gamma^{\mu}\partial_{\mu}\psi(x) - m\overline{\psi}(x)\psi(x)\Big)
[/tex]
in terms of the Fourier coefficients of psi. The answer is
[tex]
\int\frac{d^3p}{(2\pi)^3}\Big(\textrm{Re}(i\psi_p^{\dagger}\partial_0 \psi_p) - \boldsymbol{p}\cdot(\overline{\psi}_p\boldsymbol{\gamma}\psi_p) - m\overline{\psi}_p \psi_p\Big)
[/tex]
So the fermion analogy to the harmonic oscillator is
[tex]
L = \textrm{Im}(\dot{z}^{\dagger} z) - \boldsymbol{a}\cdot(\overline{z}\boldsymbol{\gamma} z) - b\overline{z} z,\quad\quad\quad z(t)\in\mathbb{C}^4
[/tex]
The classical behavior of this system is straightforward to solve. It is more complicated than the simple example in the beginning of the post, but it has the same basic properties, and in particular the EOM contains only first order time derivatives.
So ultimately, I would like to understand how to quantize this fermion oscillator, but right now I'm more interested in the simpler example, because the main difficulty is already present there.
Already known:
There is no need to preach me that it is sufficient to take the harmonic oscillator, and replace the commutation relations of the operators by anti-commutation relations. I am fully aware that this is how Dirac field is usually made to work. But besides the abstract properties of the raising and lowering operators [itex]a^{\dagger}[/itex] and [itex]a[/itex] of the harmonic oscillator, these operators also have very explicit expressions in terms of the operators [itex]x[/itex] and [itex]-i\partial_x[/itex]. I would like to have something similarly explicit for the fermion oscillator too.
How do you quantize a two dimensional system defined by the Lagrange's function
[tex]
L=\dot{x}y - x\dot{y} - x^2 - y^2 ?
[/tex]
This is a non-trivial task, because the system has some pathology. Classically the equations of motion are
[tex]
\dot{x}(t) = y(t)
[/tex]
[tex]
\dot{y}(t) = -x(t),
[/tex]
and for arbitrary initial configuration x(0), y(0), the solution is
[tex]
\left[\begin{array}{c}
x(t) \\ y(t) \\
\end{array}\right]
= \left[\begin{array}{rr}
\cos t & \sin t \\
-\sin t & \cos t \\
\end{array}\right]
\left[\begin{array}{c}
x(0) \\ y(0) \\
\end{array}\right]
[/tex]
Alternatively, the L, EOM, and solution can be written more compactly with complex numbers:
[tex]
L=\textrm{Im}(\dot{z}^* z) - |z|^2
[/tex]
[tex]
\dot{z} = -iz
[/tex]
[tex]
z(t)=e^{-it}z(0)
[/tex]
Usually the equations of motion define the second time derivatives of the coordinates, so that both coordinates and velocities are needed for unique solution. With this system velocities are not independent of the coordinates, and as consequence the usual quantization procedure doesn't really work.
An attempt with the Schrödinger's equation and Hamiltonian:
The Hamiltonian can be solved to be
[tex]
H=\frac{\partial L}{\partial\dot{x}}\dot{x} + \frac{\partial L}{\partial\dot{y}}\dot{y} - L = x^2 + y^2 = |z|^2
[/tex]
This looks pretty strange Hamiltonian, but it is in fact one of the most obviously conserving quantities in the system, so it could be considered some kind of energy. The SE is then
[tex]
i\partial_t\Psi(t,z) = |z|^2\Psi(t,z),
[/tex]
and solutions are
[tex]
\Psi(t,z) = e^{-i|z|^2 t}f(z).
[/tex]
Clearly something is wrong with this, because the quantum mechanical solutions do not give the classical behavior on the classical limit. The problem is that the canonical momenta are
[tex]
p_x = \frac{\partial L}{\partial\dot{x}} = y
\quad\quad\quad
p_y = \frac{\partial L}{\partial\dot{y}} = -x
[/tex]
so when we should leave parameters x and y untouched, and substitute [itex]p_x\to -i\partial_x[/itex] and [itex]p_y\to -i\partial_y[/itex], the coordinates and momenta get confused. I don't really know where those derivative operators should be put.
An attempt with path integrals and action:
Suppose the system goes from z' to z, in time [itex]t_1-t_0[/itex]. We can parametrize a path
[tex]
z(t) = \frac{(t_1-t)z' + (t-t_0)z}{t_1-t_0},
[/tex]
and compute the action
[tex]
S = \int\limits_{t_0}^{t_1} L(z(t))dt = xy' - yx' - \frac{1}{3}(t_1-t_0)\big( x^2 + xx' + (x')^2 + y^2 + yy' + (y')^2\big),
[/tex]
but this cannot be used to define time evolution with
[tex]
\Psi(t+\Delta t, z) = N \int dx'\;dy'\; e^{iS}\Psi(t,z')\;+\; O(\Delta t^2)
[/tex]
as usual, because there is no
[tex]
\propto \quad\frac{1}{t_1-t_0},\quad\quad\textrm{or}\quad \frac{1}{(t_1-t_0)^2}
[/tex]
kind of terms in the S as usual. Such terms are necessary to make actions for large spatial transition in small time to become infinite, and to produce necessary oscillation in the path integral.
At this point I'm out of ideas.
Motivation
This is not an unphysical example. Even though physical systems usually have EOM containing second order time derivatives, this is not always the case, and the classical Dirac field is the most obvious counter example, since Dirac equation contains only first order derivatives. Also, any initial [itex]\psi(0,x)[/itex] alone always fixes the time evolution uniquely.
One question that stroke me already in the beginning of the studies of QFT was, that if the quantization of the Klein-Gordon field is based on the quantization of harmonic oscillators, then what is the quantization of the Dirac's field based on? This question is easily answered by writing the Lagrangian
[tex]
L = \int d^3x\; \textrm{Re}\Big(i\overline{\psi}(x)\gamma^{\mu}\partial_{\mu}\psi(x) - m\overline{\psi}(x)\psi(x)\Big)
[/tex]
in terms of the Fourier coefficients of psi. The answer is
[tex]
\int\frac{d^3p}{(2\pi)^3}\Big(\textrm{Re}(i\psi_p^{\dagger}\partial_0 \psi_p) - \boldsymbol{p}\cdot(\overline{\psi}_p\boldsymbol{\gamma}\psi_p) - m\overline{\psi}_p \psi_p\Big)
[/tex]
So the fermion analogy to the harmonic oscillator is
[tex]
L = \textrm{Im}(\dot{z}^{\dagger} z) - \boldsymbol{a}\cdot(\overline{z}\boldsymbol{\gamma} z) - b\overline{z} z,\quad\quad\quad z(t)\in\mathbb{C}^4
[/tex]
The classical behavior of this system is straightforward to solve. It is more complicated than the simple example in the beginning of the post, but it has the same basic properties, and in particular the EOM contains only first order time derivatives.
So ultimately, I would like to understand how to quantize this fermion oscillator, but right now I'm more interested in the simpler example, because the main difficulty is already present there.
Already known:
There is no need to preach me that it is sufficient to take the harmonic oscillator, and replace the commutation relations of the operators by anti-commutation relations. I am fully aware that this is how Dirac field is usually made to work. But besides the abstract properties of the raising and lowering operators [itex]a^{\dagger}[/itex] and [itex]a[/itex] of the harmonic oscillator, these operators also have very explicit expressions in terms of the operators [itex]x[/itex] and [itex]-i\partial_x[/itex]. I would like to have something similarly explicit for the fermion oscillator too.
Last edited: