Astronomy Cast Ep. 249: Schrödinger’s Cat

Neat conversation. I’ll donate a Cat!!! Her name is Sadie and she’s not very friendly.

The Schrodinger cat is a way of illustrating the question on where the cut-off in a measurement takes place. This was elucidated by Heisenberg in a letter to Bohr. Bohr advanced the idea that nature existed in a duality between a quantum domain and classical reality. Bohr stated that a quantum system must be measured by a classical system which absorbs a superposition in the so called collapse of a wave function. A quantum system exists as a Fourier summation of possible wave-states

|?> = sum_n c_n|n>

for |?> the state vector and |n> a basis state in some choice of bases. The collapse of the wave function is then a case where |?> — > |m> which occurs with a probability P_m = c*_mc_m. The c_n’s are amplitudes which have various frequencies and this summation is similar to the harmonics one might detect in listening to a musical note. Different superpositions lead to different probability distributions of outcomes, and we might compare this to how a violin sounds differently from a trumpet when they play the same note. The various Fourier modes then give probability distributions and the unity of the state vector is

1 = = sum_{mn} c_mc_n

where the basis vectors are perpendicular to each other = 1 only if m = n and so we get the probability rule 1 = sum_nc*_nc_n = sum_nP_n. The act of measurement has the effect of adjusting the probability distributions, so that if the mth state is detected the probability right after that detection is P_m = 1. Heisenberg noted something funny here, for where is the cut-off for a classical system that acts as an objective measurement system?

I indicate below a more complete discussion of the measurement process, for those who are physics geeks who get into linear algebra and quantum states. Pamela used the term entanglement, and that is just exactly what happens. The superposition of states is replaced by an entanglement between the system measured and some quantum state associated with an apparatus needle. The question is then what measures the needle? So there must be an entanglement of the needle state with another state, say a heavier needle state. The heavier needle state means the uncertainty in the energy of the heavy needle state ?E < ?, but we know this can only exist in some idealization as a limit. Then in some hierarchy of needle states of increasing mass you can end up with entanglement of detector state which is huge, say a cat that can exist in a live and dead state. So then we need an observer of the cat. So Wigner’s friend is in a box which surrounds the box, say wearing an appropriate gas mask, can open the cat box to observe the cat and record the outcome: Dead or Alive. However, Wigner sitting outside his friend’s box can’t know whether his friend has written Alive or Dead, so he interprets his friend’s state, or the state of his notebook, as similar to the cat. This would be contrary to what his friend might at that moment have observed and be thinking. This is a case of logic where A or B may be a true statement, but correspondingly A is not true and B is not true.

We then have a funny situation, where the observed state of a quantum system is ultimately subjective. By subjective I am using the word in the meaning of Bayesian statistics. Quantum mechanics is sort of the ultimate Bayesian system. The measurement cut-off occurs on the basis of the prior probability estimate used by an observer. A sequence of measurements, say in the above hierarchy, is similar to a Bayes’ rule regression where priors are refined. Where this regression cuts off is not determined by nature, but by the choice of experiment the observer makes.

This has interesting connections to black holes and the relationship between quantum mechanics and general relativity. In particular it concerns aspects of semi-classical gravity with

R_{ab} – 1/2Rg_{ab} =

for this really obtains in the case where the expectation is evaluated on some choice of a frame. The is a quantum expectation for the matter-field operators which act as a source of gravitation. This expectation is expressed according to some basis choice used in a measurement. This basis choice is set on a particular reference frame in spacetime, where on that frame a measurement is performed, giving the right hand side , whereas on the left we have a covariant expression of curvature. So something is rather amiss with this, for the left hand side is frame independent, where the right hand side is represented in some basis and frame set by an observer. The right hand side is the appropriate in some pseudo-tensor form, and to make this consistent we have to include all possible measurements on all possible frames. So there is then some summation over all decoherent histories with pseudo-tensor outcomes. The most important dichotomy which happens is with respect to the quantum physics an observer outside a black hole measures verses an observer interior to a black hole. The two observers detect quantum fields or strings in complementary basis states. Even though these two outcomes will be completely different, there is nothing in nature herself which tells us which is correct.

I have to write something concerning the question posed by the British man in San Francisco about the Wheeler Delayed Choice Experiment (WDCE) on cosmic distances. The idea is that large elliptic galaxies act as beam splitters. Photons with the same wavelength reaching a detector can exhibit the Hanbury Brown-Twiss effect, whereby they fall into the same state. So if we observe a photon coming from a source it can have a superposition of having traveled along one side of the beam splitter or the other. If we use radio telescopes with a narrow frequency band pass so the momentum uncertainty is ?p ~ 0 then by the Heisenberg uncertainty principle ?p?x = ? (minimal uncertainty) the spread in x is very large ?x >> 0. In this way the difference in distance between the two “arms” of the galactic beam splitter can be made smaller than the uncertainty. In this set up it is then possible to perform certain experiments which can select which path the photon traveled in a WDCE. This is even if the photon traversed billions of light years from a galaxy billions of light years away.

As I indicated above, I give an account of the measurement process for the two slit experiment. Suppose you have a two state system of some sort you are looking at. In the case of a double slit there are two possible superposed states |+> and |-> for quanta occurring at the + slit and the – slit. These states then have some expansion according to their appearance on the screen, so we have

|+> = sum_n a_n|n>, |-> = sum_n b_n|n>.

It the slits are opened then the state which occurs on the screen is the summation of these two, so then

|?> = 1/sqrt{2}(|+> + |->) = sum_n(a_n + b_n)|n>,

where I have absorbed the normalization into the amplitudes. So we then compute

= 1 = sum_{mn}

= sum_n(|a_n|^2 + |b_n|^2 + a_nb^*_n + a_n^*b_n)|n>.

If the amplitudes are a_n = e^{inkx} b_n = e^{ink’x} then

a_nb^*_n + a_n^*b_n = 2cos(n(k – k’)x)

which is the interference pattern we expect.

Now let us suppose we couple this to some auxiliary system. Suppose there is a little detector at one of the slits, so that it goes into the “up state” |u> if the quanta passes through the upper + slit. So we have the up and down slit states as

|+,u> = sum_n a_n|n,u>, |-,d> = sum_n b_n|n,d>,

and now the total state is the sum of these

|?> => 1/sqrt{2}(|+,u> + |-,d>) = sum_n(a_n|u> + b_n|d>)|n>,

where now we have two orthogonal conditions due to the fact we have entangled the double slit states with some other state. The result of the entanglement means we now have that the cross terms a_nb^*_n + a_n^*b_n are multipled by and which are orthogonal and zero. The entanglement condition has destroyed the interference pattern, where one strange aspect of quantum mechanics, the superposition of two states is taken up by in an entanglement with another pair of states.

This is the so called collapse of the wave function, and is the source of all this trouble with quantum mechanics. Now for the first situation the density matrix of the two slit experiment is a matrix with probabilities which can equal 1/4 in all the entries for the particle reaching the screen center. By contrast the density matrix reduced to the slit states have entries 1/2 when the system is entangled with this auxiliary spin state. The Schrodinger cat argument involves a hierarchy of measurement “needle states” which measure this auxiliary spin state, and a measurer of that needle state and so forth up to a large scale system — say the brain of a physicist.

LC