unitarily inequivalent representations

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In ordinary quantum mechanics of systems with finitely many degrees of freedom, the "choice problem", namely choosing a particular representation of the canonical commutation relations, is resolved by the Stone–von Neumann theorem which states that any representation is essentially unique, up to unitary transformations. This is not so for systems with infinitely many degrees of freedom (e.g., Quantum field theories), and the statement is known as Haag's theorem[1]

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Example: Lattice of spins

Consider a cubic lattice with a particle with spin lying on each vertex. The spin of each vertex is measured by the observables \sigma_x(n)\,, \sigma_y(n)\,, \sigma_z(n)\,, which are the Pauli matrices and where n\, labels the lattice site[2]. In the presence of a magnetic field in the z\, direction, the ground state is denoted by \left|0\right\rangle = \left|\uparrow\uparrow\uparrow...\right\rangle\,. Suppose we choose a different coordinate system by rotating our axes about the y\,-axis. This is done by the unitary operator U = e^{-i\frac{\theta}{2}\sum_{m=1}^N \sigma_y(m)}\,. Then

\sigma_i'(n) = U \sigma_i(n) U^{-1}\,.

What is the overlap between the original state \left|0\right\rangle\, and the rotated state

\left|\theta\right\rangle = e^{-i\frac{\theta}{2}\sum_{m=1}^N \sigma_y(m)} \left|0\right\rangle\,?

Clearly,

\left|\theta\right\rangle = \cos^N\frac{\theta}{2} \left|0\right\rangle + ...\,,

therefore as N\to\infty\,, the overlap with the original state becomes zero. Moreover, since the coefficient of the identity in U\, is \cos^N\frac{\theta}{2} \to 0\,, the overlap of any vector with its rotated counterpart is zero. Finally, note that

e^{-i\frac{\theta}{2}\sum_{m=1}^N \sigma_y(m)} = \prod_{m=1}^N \left(\cos\frac{\theta}{2} + i \sigma_y(m) \sin\frac{\theta}{2} \right)\,.

The action of U\, on a physical state \left|\psi\right\rangle\, (obtained by finite application of \sigma_{-}\, on \left|0\right\rangle\,), will involve a sum of N\, states. Each state will have s\, factors of \cos\frac{\theta}{2}\, and N-s\, factors of i \sin\frac{\theta}{2}\sigma_y\, where s \in [0,N]\,. Either s\, is finite, in which case there are infinitely many factors of \sigma_y\, as N\to\infty\,, (which corresponds to an unphysical state), or N - s\, remains finite as N \to \infty\,, i.e., it overlaps with physical states, but then we still have to contend with the factor \cos^s\frac{\theta}{2} \sin^{N-s}\frac{\theta}{2}\to 0\, as s \to \infty\,. In either case, the overlap with a physical state for that particular term tends towards zero as N \to \infty\,. Thus the sets of rotated and unrotated physical states have no states in common. They are unitarily inequivalent and the operator U\, is not well defined as N \to \infty\,.

Example: Free scalar field

Suppose we have a free scalar field diagonalized by the creation and annihilation operators a^\dagger_k\, and a_k\, where \left[a_k, a^\dagger_{k'}\right] = \delta_{k,k'}\, and for the moment k\, labels a discrete momentum variable. Now suppose we choose a different representation, namely

b_k = \cosh \varepsilon \,a_k + \sinh \varepsilon\, a^\dagger_k\,,
b^\dagger_k = \sinh \varepsilon \,a_k + \cosh \varepsilon \,a^\dagger_k\,,

which has the same commutation relation \left[b_k, b^\dagger_{k'}\right] = \delta_{k,k'}\,, since \cosh^2\varepsilon - \sinh^2\varepsilon = 1\,. This is a Bogoliubov transformation. Infinitesimally,

\delta a_k = \varepsilon\, a^\dagger_k\,,
\delta a^\dagger_k = \varepsilon \,a_k \,,

which is generated by the operator

T = \frac{1}{2} \sum_k \left( a^\dagger_k a^\dagger_k - a_k a_k \right)\,,

i.e.,\delta a_k = i \varepsilon \left[T, a_k\right]\, and \delta a^\dagger_k = i \varepsilon \left[T, a^\dagger_k\right] \,. Consider the effect of the unitary transformation on a (normalized) state \left|\Psi\right\rangle_a\, in the Hilbert space H_a\,:

\left|\Psi\right\rangle_b = e^{i\varepsilon T} \left|\Psi\right\rangle_a\,.

Because it is unitary, the norm is unchanged.

For simplicity, lets concentrate on the effect on the vacuum, considering only one momentum k\,.

e^{i\varepsilon T_k} \left|0\right\rangle_a\, = \left(1 + i\varepsilon T - \varepsilon^2 T^2 + ... \right)\left|0\right\rangle_a\,
= \left|0\right\rangle_a + i\varepsilon \frac{\sqrt{2!}}{2} \left|2_k\right\rangle_a - \varepsilon^2 \frac{1}{2^2} \left( \sqrt{4!} \left|4_k\right\rangle_a + 2\cdot 1\sqrt{2!}\left|0\right\rangle_a \right) + ... \,
= \left(1 -\frac{\varepsilon^2}{\sqrt{2}}+... \right)\left|0\right\rangle_a + ... \,

For small enough \varepsilon\,, the factor multiplying \left|0\right\rangle_a\, is less than 1\,. If we consider the effect of the full transformation (i.e., all momenta), then

e^{i\varepsilon T_k} \left|0\right\rangle_a \to \left(1 -\frac{\varepsilon^2}{\sqrt{2}}+... \right)^\infty \left|0\right\rangle_a + ... \,,

so that if we allow for infinitely many degrees of freedom, {}_a\left\langle 0| 0\right\rangle_b = 0\,. We can apply the same reasoning as in the case of a lattice of spins:

U = \prod_k e^{i \varepsilon T^k}\,.
U = \prod_k \left(1 + i \varepsilon T^k + ...\right)\,,

so upon the action on a physical state, a general term in the sum will contain s\, factors of \left(1 -\Delta_k\right)\, and N - s\, actions that create or destroy particles. In order for the state to remain normalized, we can see that \Delta_k\, doesn't go to zero as N \to \infty\, (otherwise U_k = (1-\Delta_k) + ...\, simply equals 1\,). For finite s\, the action creates infinitely many particles, while for infinite s\, (finitely many particles), the action of \prod_k \left(1 -\Delta_k \right)\, is zero.

Squeezed states

From our knowledge of squeezed states we have an explicit relation between the vacua of the two coordinate systems:

\left|0\right\rangle_b = e^{-\frac{1}{2}\tanh\varepsilon\sum_p(a^{\dagger}_p)^2}\left|0\right\rangle_a\,,

i.e.,

\left( \cosh \varepsilon \,a_k + \sinh \varepsilon\, a^\dagger_k\right) e^{-\frac{1}{2}\tanh\varepsilon\sum_p(a^{\dagger}_p)^2}\left|0\right\rangle_a = 0\,\,\forall k\,.

Consistent limits

References

  1. ↑ Haag, R: On quantum field theories, Matematisk-fysiske Meddelelser, 29, 12 (1955).
  2. ↑ C. Itzykson, J. Zuber (2006). Quantum Field Theory. Dover, New York. ISBN 978-0486445687. 
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