Polyhedra and Groups
The central idea of this article is a direct, geometric method to multiply/compose permutations, by using polygons and regular polyhedra.
I begin with a short review of the properties of regular polyhedra and permutation groups, as well as of the relations between these two areas. These relations led me to the geometric method of multiplying permutations.
Regular polyhedra
The regular polyhedra, also known as platonic solids, are represented below:
Inscribing one regular polyhedron in another
There are many ways to inscribe one regular polyhedron in another. One useful case is that when each vertex of the first polyhedron lies at the center of a face of the second, and each face of the second polyhedron contains one vertex of the first.
The cube can be inscribed in this way in the octahedron, but also reciprocally. The same reciprocity holds between the icosahedron and the dodecahedron. Such polyhedra are said to be dual. The tetrahedron is self-dual.
The following ways to inscribe regular polyhedra will be useful too: one or two tetrahedra in a cube, and cube in a dodecahedron.
You can permute the set {1, 2} in two ways: (1, 2) and (2, 1). What about the set {1, 2, 3}? It is easy to see that we have six possibilities: (1, 2, 3), (2, 3, 1), (3, 1, 2), (2, 1, 3), (3, 2, 1), (1, 3, 2). The result is the same, for any sets of three objects, not only numbers. The number of permutations depends only on the number of elements of the set, and not on their nature. For a set with n elements, this number is 1·2·...·n, and is named the factorial n! of the number n.
An ordered set of the first n positive integers can also be used to express the permutation, as a specific operation to be applied to another ordered set. For example, (2, 1) shows that the elements of an ordered set of two elements are reverted. (2, 3, 1) shows that an ordered set of three, for example (1, 2, 3), changes its order to (2, 3, 1). Such transforms are named permutations of the set {1, 2, ... , n}.
We can multiply two permutations, which means that we apply them successively to the ordered set. All the permutations of a set with n elements forms a grup, because the operation of multiplication of permutations is associative, has a neutral element (which is the identical permutation (1, 2, ... , n), leaving the order unchanged), and every permutation has an inverse which cancels its effect. The group of permutations of the set {1, 2, ... , n} is named the symmetric group of degree n, denoted by $S_n$, and having n! elements.
A transposition is a permutation which interchanges only two elements. For example, (2, 1, 3), (3, 2, 1), (1, 3, 2) are transpositions of the group $S_3$. Any permutation can be decomposed as a product of transpositions in a non unique way. Yet, there is something independent of the way we decompose the permutation as a product of transpositions. If the number of transpositions in such a decomposition is even, it will be even for any other decomposition of the same permutation, and we name such a permutation even. Otherwise, we call it odd. The set of even permutations of a set with n elements forms a subgroup of $S_n$, named the alternating group, denoted by $A_n$, and having n!/2 elements.
The symmetries of a regular polyhedron
By rotating the regular tetrahedron around one of its heights with 120º or 240º, this one remains unchanged. We say that the regular tetrahedron is unchanged by these transformations. We can rotate a regular polyhedron so that, after this transformation, it occupies exactly the same position, but having the faces not necessarily in the same positions. Also, they admit symmetry planes. In fact, this is the idea about the regular polyhedra – their rich symmetry.
The transformations leaving unchanged a polyhedron are named symmetry transformations. One can multiply the transformations. Because each symmetry transformation interchanges the faces of the polyhedron, we can associate to the transformation a permutation from $S_n$, n being the number of faces of the polyhedron. The symmetry transformations leaving invariant a polyhedron form a subgroup of $S_n$, named the automorphism group of the polyhedron.
How many symmetry transformations have each of the regular polyhedra? One easy method to count them is the following. Choose a face a. After a transformation it will take the place of another face a'. Since the polyhedron has n faces, we have n possibilities. Let's choose a face b, neighbor to the first face before transformation, a. After the transformation, b goes into a face b', neighbor to the face a'. Each face has the same number of edges k. The face b' can be one of the k faces neighbor to a'. Therefore, we have nk possibilities. These transforms are the rotations of the polyhedron. But the face a is a polygon, therefore it has two sides. Consequently, when we move the face a in a', this can flip. In this case, the transformation is no longer a rotation. It can no longer be obtain simply by moving the polyhedron, but by taking its mirror image.
Thus, among all transformations of the polyhedron, a special subgroup is formed by the nk rotations, but the total number of transformations is 2nk. Both these groups are subgroups of $S_n$. For the regular tetrahedron we obtain 2·4·3 = 24 automorphisms, from which the rotations are 4·3 = 12. The cube's automorphisms group contains 2·6·4 = 48 automorfisms, from which 6·4 = 24 are rotations. The regular octahedron has 2·8·3 = 48 automorfisms, from which 8·3 = 24 rotations, like the cube. Both the icosahedron and the dodecahedron have 2·4·3 = 24 automorfisms, 4·3 = 12 rotations.
We see that the dual polyhedra have the same symmetries.
The symmetries of regular polyhedra and the permutations
Let's label the vertices of a regular tetrahedron with the numbers {1, 2, 3, 4}:
At a symmetry transformation, the vertex 1 can go in any of the four vertices. The vertices 2, 3 and 4 are all neighbor with the vertex 1, and they will remain so after the transformation too. Their order around the vertex 1 is preserved in the case of rotations, otherwise it is reverted. Therefore, the vertices' permutation is even if and only if the transformation is a rotation. The regular tetrahedron having 24 automorphisms, they coincide with the elements of the group $S_4$. The rotations correspond to the elements of the alternating group $A_4$.
Let us now label the cube's vertices such that the ends of each large diagonal have identical labels from the set {1, 2, 3, 4}, like below:
Let's choose one of the two regular tetrahedra inscribed in the cube. Its vertices are labeled with the numbers {1, 2, 3, 4}. A rotation of the cube rotates also the tetrahedron. Each cube rotation which let the tetrahedron in place corresponds to a rotation of the tetrahedron, and it is an even permutation. The even permutations of the cube's labels correspond to transformations which interchange the two tetrahedra. Thus, the cube's rotations correspond to the symmetric group $S_4$.
By labeling the faces of the regular octahedron, we obtain that it has the same symmetry groups as the cube (being its dual).
To obtain the symmetry group of the regular dodecahedron, let's label its edges with the numbers {1, 2, 3, 4, 5}, like in the image:
For each edge of the dodecahedron, we take the four adjacent edges, and the other vertices of these edges form a square (hint: the four edges are diagonals in identical regular pentagons). These edges form five cubes. Each of the five cubes highlights six of the dodecahedron's edges. It is easy to see that for each face we obtain a different ordering for the labels. Each rotation of the dodecahedron will take the face labeled by the ordered set (1, 2, 3, 4, 5) (we count starting with the vertex 1) in another face, also labeled with a permutation of the five numbers, so that 1 goes to one number, 2 to another etc. Thus, to the permutation (1, 2, 3, 4, 5) we can associate, as a result of the rotation, another permutation. Because we limit ourselves to rotations, we can choose one orientation (for example clockwise). The rotations will keep this orientation. Because we can start from any vertex of a face when we read the permutation, we will have five permutations for each face. This way, each corner of a face represents a permutation. We can check that these permutations are always even. Each dodecahedron's rotation corresponds to an even permutation of the face labeled by (1, 2, 3, 4, 5). The dodecahedron's rotations group is isomorphic with the alternating group $A_5$.
The regular icosahedron being dual to the regular dodecahedron, its 30 edges can be labeled like the ones of the dodecahedron. In this case, we will use cubes with the vertices in the centers of the icosahedron's faces.
Group operations with polygons
"Calculator" for the Klein group
The Klein's group has four elements {E, A, B, C}, and its multiplication is given by the label:
This group is isomorphic with the automorphism group of a rectangle. We can construct a "calculator" for multiplying elements of the Klein group by using to identical cards:
The first card will be used as witness card. For obtaining all the multiplications with the element B, hence the "multiplication with B table". For doing this, we rotate the second card (the result card) so that the element B is moved in the position corresponding to the neutral element E of the witness card:
Now, to obtain the product of any element X from the group with B, we just read from the result card the element from the position corresponding to the position of X in the witness card. For example, to see the result of the operation C·B, we look for the element C in the witness card. It is in the lower right corner. We look in the same position in the result card. The corresponding element is A. We can check in the multiplicative table that indeed C·B = A.
Calculator for the groups $A_3$ and $S_3$
We start with the group $A_3$ of the even permutations of a set with 3 elements. Its elements are the permutations (1, 2, 3), (2, 3, 1), (3, 1, 2). We construct a card shaped as an equilateral triangle and we label its vertices with the numbers 1, 2 and 3:
The rule is: we label the vertices with the numbers {1, 2, 3} clockwise, and the edges with the permutations given by the order in which we met the vertices by starting from that edge and go clockwise.
We see that the rotations with 120º, 240º or 360º preserve this triangle. To find the multiplications of the elements of the $A_3$ with one element of the group, say (3, 1, 2), we keep the witness card in the normal position, and rotate the result card such that the permutation (3, 1, 2) corresponds to the position of the identical permutation (1, 2, 3) from the witness card. In order to find the result of the multiplication of a permutation with the permutation (3, 1, 2), we search on the witness card the position of the desired permutation, and on the result card we just read the permutation on the corresponding position. For example, to find the result of multiplying the permutations (3, 1, 2) and (2, 3, 1), we look on the result card for the permutation corresponding to the position occupied by the permutation (2, 3, 1) on the witness card. We see that the result is the permutation (1, 2, 3):
The group $S_3$ contains, in addition to the even permutations (1, 2, 3), (2, 3, 1), (3, 1, 2) from the group $A_3$, the odd permutations (2, 1, 3), (3, 2, 1), (1, 3, 2). This is why we will allow, besides the rotations maintaining the triangle in plane, transformations obtainable by lifting the triangle from the table and flipping it. On its back face we will mark the odd permutations:
Calculator for the groups $C_n$ and $D_n$
A finite group such that all its elements can be obtained by multiplying one element (named generator of the group) with itself, is named cyclic group. We denote the cyclic group with n elements by $C_n$. The cyclic group $C_n$ is isomorphic with the group of integers modulo n, $\mathbb{Z}_n$. The cyclic group C3 is isomorphic with the alternating group $A_3$, but this doesn't hold for n > 3. We can consider the group $C_n$ as representing the plane rotations of a regular polygon with n edges. If we allow mirror symmetries, obtained by taking the polygon outside the plane and flipping, the number of the possible symmetries doubles, and their group is named the dihedral group, $D_n$. We observe from the definition that for n = 3 the dihedral group is isomorphic to the permutation group of a set of three elements: $D_3$ ~ $S_3$, but, as for the cyclic group, we can't generalize for n > 3.
The calculators for the cyclic and dihedral groups can be made from regular polyhedra. They can be used similarly to those described for the groups $A_3$ and $S_3$, with a witness polygon and a result polygon.
Polyhedral calculators of permutations
Calculator for the group $S_4$
Let's consider again the cube with vertices labeled like this:
Each face has four edges, and we label each of them, on that face, with the permutations obtained by reading the vertices' labels, starting from that edge and walking clockwise. We obtain a cube labeled on each side of each edge with permutations. The reader can construct her own cube by printing this image:
After assembling it in 3D, it will look like this:
For finding all the multiplications of the permutations from $S_4$ with a particular permutation, say (4132), we put the witness cube with the face containing the identical permutation (1234) in front, such that the identical permutation is below:
Then we put the resulting cube in a similar position, only that on the position of the identical permutation we put the permutation (4132):
Let's suppose we want the result of the multiplication of the permutation (4132) with another permutation from $S_4$, for example (3124). We look in the witness cube the position of the permutation (3124), and in the result cube we read the permutation from the corresponding position. Because the permutation (3124) occurs in the witness cube on the left edge of the right face, we read the permutation from the left edge of the right face of the resulting cube. This is (3412).
Once we position the cubes, we can simply read all the results of the multiplications with the chosen permutation (4132) with any element of $S_4$ on the result cube.
The reader is invited to study the symmetries of the permutations written on the cube.
For example, two permutations associated to the same edge differ by a transposition between the elements labeling the edge's ends. The permutation (4123) is neighbor of the permutation (3124) – they are on the same edge of the cube. Their common edge has the vertices labeled with the numbers 4 and 3. These numbers are transposed in one of the two permutations to obtain the other. In fact, any two permutations associated to the same edge differ by inverting the positions 1 and 4.
Another interesting property of this cube is that, applying a rotation, we obtain on the initial position of the identical permutation a permutation showing how the four large diagonals of the cube were permuted.
Calculator for the group $A_5$
As we remember, the group of rotations of the dodecahedron is isomorphic with the alternating group $A_5$. We can label the dodecahedron by using even permutations of the set {1, 2, 3, 4, 5}, like this:
By printing it and gluing the edges, you can construct a dodecahedron. By constructing two identical such dodecahedra, we can use them to multiply permutations from the alternating group $A_5$, in a similar way we used two cubes to multiply elements of $S_4$.
Posted by Cristi Stoica at 12:47 AM
Labels: Fun, Puzzles, Symmetry
Smooth Quantum Mechanics: 0. Main Article
Smooth Quantum Mechanics
Quantum Mechanics without discontinuities in time evolution
I show that the apparent wave function collapse can take place smoothly, without discontinuities. The projections on the observable's eigenspaces can be obtained by delayed initial conditions, imposed to the smooth time evolution of the observed system entangled with the measurement device used for the preparation. Since the quantum state of this device is not entirely available to the observer, its unknown degrees of freedom inject, by the means of entanglement, an apparent randomness in the observed system, leading to a probabilistic behavior. By using this mechanism, we can construct a Smooth Quantum Mechanics (SQM), without the need of discontinuities in time evolution. Therefore, the probabilities occur because not all the involved systems have determined quantum states. The evolution is deterministic, but for an observer who has access only to an incomplete set of initial conditions, it appears to be indeterministic.
The problem of discontinuities in Quantum Mechanics
The time evolution of a Quantum System
A quantum system which is in a pure state, and not entangled with another system, evolves according to the Schrödinger equation:
$$\Bigg\{
\begin{array}{ll}
i\hbar\frac{\partial}{\partial t}|\psi(t)\rangle=H(t)|\psi(t)\rangle \\
|\psi(t_0)\rangle=|\psi_0\rangle,
where $|\psi(t)\rangle\in\mathcal{S}$ is a state vector from the state space $\mathcal{S}$, and $H(t)$ is the Hamiltonian, usually a hermitian operator on $\mathcal{S}$. If the quantum system is closed, then $H$ is time independent, but in general, it is time dependent, because of the interactions with other systems. For more general interactions, the observed system can become entangled with other systems, and its state will no longer be pure. In this case, as well as in the case when we don't know the initial data, but rather a probability distribution, we represent the state by a density operator $\rho$ on $\mathcal{S}$. For these situations we will employ, instead of the Schrödinger's equation, the Liouville - von Neumann equation:
i\hbar\frac{\partial\rho(t)}{\partial t}=[H(t),\rho(t)] \\
\rho(t_0)=\rho_0.
The measurement problem
When a measurement is performed to a quantum system, the system is found to be in an eigenstate of the observable. There are two main problems raised by this fact. First: why only eigenstates of the observables are obtained as outcomes of the observations? Why don't we observe superpositions of such eigenstates, like a $|\textrm{dead}\rangle+|\textrm{alive}\rangle$ Schrödinger cat? This is the main problem of the measurement, which, in this article, will be accepted as it is, without offering an explanation. The second problem is the following. Knowing the state of a quantum system, and assuming that a measurement will find the system in an eigenstate of the observable, it seems like a discontinuous jump happens. How is this happening? Is this jump really discontinuous? What is its nature? In this article, I propose a solution to this discontinuity problem in Quantum Mechanics.
One-measurement
The Schrödinger's equation, as well as the Liouville - von Neumann equation, are PDE equations. Each solution can be uniquely specified by an initial condition. The initial condition is obtained by performing a measurement at an initial moment $t_0$.
Let's consider that we measure the polarization of a photon, at the instant $t_0$. The obtained result will determine the photon's polarization both for moments $t$ with $t$<$t_0$, and with $t>t_0$. The last statement needs some argumentation. A measurement is in general required to produce a minimal disturbance of the system; it is required to determine the system's state, and to let the system in the state it was found. At the quantum level, this is sometimes difficult. The outcome of a measurement depends on the observable we choose to measure. For example, by choosing a direction along which we measure the spin of an electron, we determine only two possible eigenstates, out of an infinity. In the case of the photon, we can impose the left or right circular polarization with the help of a wave plate. This is an initial condition imposed to the photon. Consider a pair of entangled photons, being obtained from a parametric down conversion, such that they have linear polarizations along directions which are orthogonal to each other. A polarization imposed to one of them, applies also to the other one. This mechanism is used in the delayed choice quantum eraser experiment [SMKYKS00]. In this experiment, the polarization of one photon determines if we have interference or not. We can choose the polarization by imposing to the other photon a polarization, and we can do this after the other photon hit the screen. The polarization imposed to one of the two photons, is revealed to apply in past, until the pair of the two photons has been emitted, so that the second photon has the correct polarization. This shows that we can impose a polarization at $t_0$, which applies for instants of time before and after $t_0$.
The classical view is that the measurement only revealed the state of the system, and the solution of the evolution equation preexisted to the measurement. On the other hand, in Quantum Mechanics, we can choose what observable to measure, thus, we can choose the set of admitted eigenstates. So, if the solution we detected by measurement preexisted, it did this in a way that anticipated our choice of the observables. This choice can be performed with a delay, to make sure that it doesn't affect in a causal way the observed system. This was emphasized by Wheeler [Whe77,Whe78,WZ83] when he revived (It seems that similar suggestions were made before by Weiszäcker [Wei31] and Bohr [Sch49].) the idea of delayed-choice experiments. In the case of the electron spin, when we choose the direction to measure the spin, we let available only two possible eigenstates for the spin. Had we choose a different direction, the eigenstates would be different. So, our choice limited the possible outcomes. And when we measure the spin, we determine not only what spin the electron has at $t_0$, but also for previous $t$, as we can see from entanglement situations like the one pointed by Einstein, Podolsky, and Rosen [EPR35] (in Bohm's version [Bohm51]).
We can conclude that one measurement determines uniquely the state at $t_0$, hence the solution, and this determination seems to affect the past in a weird way. We cannot say that it can change it, rather it is only an initial condition, established with a (very large) delay. Two measurements of the same system may be incompatible, and require something that looks like a wavefunction collapse. In the case of only one measurement (or when the measurement is the first one), the state prior to the measurement is not sustained by any data, so we can not talk about collapse. The one-measurement situation makes apparent that the eigenstate can be selected without involving the discontinuous wave function collapse. Two initial conditions may be incompatible, but only one cannot be, since it is the only condition. If after the first measurement we perform another, the things become more difficult, but the solution is similar, as we will see next.
Two-measurements case and the wave function collapse
Let's consider a system whose evolution is described by the Schrödinger's equation. Suppose that, after an observation at $t_0$ finds the system in an eigenstate, we perform a second observation, at the time $t_1$. If the state predicted by the evolution equation is an eigenstate of the second observable, then it will be obtained at the second measurement. If not, then the second observation cannot impose an initial condition at $t_1$, compatible with the unitary evolution governed by the Schrödinger equation.
We can see that one observation imposes an initial condition to the Schrödinger equation, choosing a solution, but a second observation either confirms the solution, or it is incompatible with it. In this case, it should not be possible to perform more than one observation of a system.
A quantum system has a wave-like behavior, described by the Schrödinger equation, or by the Liouville - von Neumann equation, and a quantum behavior, expressed by the condition to be found in an eigenstate of the observable. These two behaviors seem to be incompatible.
But we know from experience that we can perform more observations to the same quantum system. This has the appearance of a jumping from one solution of the Schrödinger equation to another one, in a discontinuous fashion.
The analysis of delayed-choice experiments suggests that, if a collapse happened, it took place in advance, during the previous interaction, possible even at $t_0$ (please refer to figure 1).
Figure 1 In a delayed-choice experiment, the reduction seems to take place in advance, anticipating the experimenter's choice of the observable.
Why discontinuities cause problems?
The acceptance that a quantum system is subject of a discontinuous wave function collapse can raise several problems. On the one hand, if we consider the observed system as being a part of a larger one (perhaps the Universe), containing the measurement device too, as a quantum subsystem, the measurement can be described by the evolution equation, and we expect that no discontinuous collapse appears. But, when we refer to the observed system only, we cannot see how the discontinuity can be avoided. We seem to have a paradox: a system evolving with discontinuities, being in the same time a subsystem of one evolving smoothly.
Another problem is that the discontinuous collapse has been postulated, but never observed directly. We don't know of any mechanism which can produce it, and we don't know when exactly it takes place. An explanation is required, since we cannot accept that it simply happens.
In Quantum Mechanics, an observable that commutes with the Hamiltonian of the system is conserved during the evolution.
But the conservation holds only as long as the system evolves governed by the Hamiltonian (Schrödinger equation or Liouville - von Neumann equation).
Since performing a measurement makes the system jump in a totally different state, it is expected that the conservation laws are broken. For example, if we measure the momentum of the system, and then measure its position, then the initial momentum is lost. If we measure again the momentum, we should expect to obtain a totally different value than the first time. We can expect that, after several measurements, the conserved quantities of the system be totally blown up.
The discontinuities are incompatible with the conservation laws, but the conservation laws don't break down as a result of measurements. Something happens always to restore them.
To make them compatible, we need to appeal to a "magical postulate":
During the state vector reductions, the conservation laws can no longer be deduced from the Hamiltonian, but they must be restored in some way or another.
The problem is that we don't know any explanation for the conservation laws, other than the time evolution described by the Schrödinger's equation and Liouville - von Neumann equation. Breaking this evolution should break the conservation laws, contrary to our experimental observations.
The quantum world is like a great illusionist, who has in his sleeves a lot of tricks that make us believing that the quantum system jumps discontinuously from time to time. But we have to remember that, in the end, there must be a logical explanation for the illusion number presented in the show, and to look for the strings.
Can discontinuities be avoided?
In the following, I will show that the apparent wave function collapse can be explained by the standard Quantum Mechanics, minus the discontinuity, as a smooth and natural phenomenon. The first ingredient comes from the discussion above, concerning a system undergoing only one measurement. A measurement fixes the initial data for a quantum system; going to a larger system, makes those initial conditions insufficient, therefore, a new measurement is allowed. The second ingredient is the entanglement with the device performing the previous measurement (which will be named preparation device).
Quantum Mechanics without discontinuities
We begin by considering the measurement from a semi-classical viewpoint: the observed system is quantum, and the preparation device is classical.
The semi-classical interaction approach
Let's consider a quantum system evolving according to the Schrödinger's equation, subject to a first measurement (the preparation) starting at the instant $t_0$ and ending at $t_0+\varepsilon$, and a second measurement at $t_1>t_0+\varepsilon$. If we consider the preparation device as being classical, its influence can be described by an interaction Hamiltonian $H_{\textrm{int}}(t)$. Thus, in the Dirac picture, the Hamiltonian is:
$$H(t)=H_0+H_{\textrm{int}}(t).
The preparation device is considered classical, this meaning that its true state, which is quantum, is unknown. There will be a large set of quantum states which, at the classical level, will look identical. This set of equivalent quantum states can be parameterized, with both discrete and continuous parameters. Let's take a smooth parameterization $u(t)$ of its continuous degrees of freedom. The interaction Hamiltonian $H_{\textrm{int}}(t)$ will depend on $u(t)$, such that $H_{\textrm{int}}(t)=H_{\textrm{int}}(t,u)$.
Each choice of the parameters $u(t)$ will lead to a state of the system at $t$ given by
$$|\psi(t,u)\rangle=U(t,t_0,u)|\psi(t_0)\rangle.
Before the introduction of the degrees of freedom parameterized by $u(t)$, there was only one possible state at $t_1$ for the observed system. Now, by varying $u$, $|\psi(t_1,u)\rangle$ also changes.
We ask the following question:
What condition should the parameters $u$ satisfy, such that all possible outcomes of any possible observation taking place at $t_1$ are reached by $|\psi(t_1,u)\rangle$?
This is a problem of Quantum Control Theory. Under some general assumptions on $u(t)$, the condition is that the Lie group associated to the Lie algebra generated by the matrices of the form $iH(t,u)$ should contain all the possible unitary transformations. If the dimension of $\mathcal{S}$ is $n<\infty$, then it is enough that the rank of the Lie algebra generated in this way to be identical to the rank of the unitary Lie algebra $\mathfrak u(n)$. This holds when there is no time limit, but in our case, the time is limited to $t_0+\varepsilon$, bringing a new restriction. On the other hand, the Born rule assigns zero probability to projections on an orthogonal state. Therefore, we don't need to obtain at $t_1$ states orthogonal to $|\psi_0\rangle$. The parameters $u(t)$ can be determined by appropriate initial conditions. Similarly to the one-measurement case, the initial conditions are determined such that the system evolves to be the appropriate eigenstate of the observable, at $t_1$. In the figure 2 we can see how the Hamiltonian can prepare the observed system to be in an eigenstate of the observable.
Figure 2 The disturbance in the evolution of the quantum system, introduced by the measurement device performing the preparation, needs to be taken into account by modifying the Hamiltonian from $H_0$ to $H(t,u)=H_0+H_{\textrm{int}}(t,u)$ for the time interval $(t_0, t_0+\varepsilon)$. This will "repair" the discontinuity presented in the figure 1.
Assuming that the observable corresponding to the measurement at $t_1$ is $O_1$, for each outcome $|\psi_{O_1,\lambda}\rangle$ of the measurement, corresponding to an eigenvalue $\lambda$, there must exist a choice $u_{O_1,\lambda}$ of the parameters $u(t)$ such that the interaction send the observed system in $|\psi_{O_1,\lambda}\rangle$. The corresponding unitary operator is $U_{O_1,\lambda}(t_1,t_0)$, so that $|\psi_{O_1,\lambda}\rangle =U_{O_1,\lambda}(t_1,t_0)|\psi_0\rangle$.
Let us consider the following example, raised by Einstein to Bohr, at the Fifth Solvay Conference (Brussels 1927). Einstein said that, in a two-slit experiment, if we measure the recoil of the wall containing the two slits, when the light passes through it, one should be able to deduce whether the photon passed through one slit or the other. As Bohr replied to him, if we detect a significant recoil, the interference pattern is destroyed.
Let's reverse a bit the reasoning, and apply it to the delayed-choice [Whe77, Whe78, WZ83, Wei31, Sch49] version of the two-slit experiment. We can decide after the photon has passed through the slit(s) whether to observe the "which way" or the "both ways" aspects. If we decide to observe the "which way" behavior, we cause the wall with the two slits to undergo a significant change of momentum (corresponding to the cases when the photon has passed through one slit or the other). If we choose to observe the interference, the change in momentum will be undefined. The wall with the two slits will get in a superposition of eigenstates of momenta.
This example shows that, indeed, the interaction with the wall with the two slits, happening in the interval $(t_0,t_0+\varepsilon)$, takes place in such a manner that the outcome of the measurement is one of those expected by the choice of the observable.
The entanglement approach
The previous analysis simplified the interaction between the preparation device and the observed system. A more general description will consider that the preparation device is quantum, not classical. In this case, its interaction with the observed system leads to an entanglement between the two. The evolution of the observed system can no longer be considered unitary: its state may go from being pure, at $t_0$, to being mixed at $t_0+\varepsilon$. Of course, the combined system made of the observed system and the preparation device, may be isolated, and undergo unitary evolution, but the observed system's state will be obtained by partial tracing the density operator of the larger system, and it will not necessarily be pure.
A correct description will use density operators to represent the state, and the Liouville - von Neumann equation, for its evolution.
Let us consider that the state of the observed quantum system is described by the density operator $\rho_q$, on the state space $\mathcal{S}_q$, and the one of the preparation device is described by a density operator $\rho_p$ on the state space $\mathcal{S}_p$. We consider that the combined system, represented by a density operator $\rho_{q,p}$ on $\mathcal{S}_q\otimes \mathcal{S}_p$, is isolated. If it is not isolated, then we complete the system with remaining systems $\rho_r$, so that we obtain an isolated system. We can consider, without loosing the generality, that the preparation device incorporates all these systems, so it will be enough to work in the state space $\mathcal{S}_q\otimes \mathcal{S}_p$. The combined system will have a unitary evolution between $t_0$ and $t_0+\varepsilon$, given by the unitary operator $U_{q,p}=U_{q,p}(t_0+\varepsilon,t_0)$:
$$\rho_{q,p}(t_0+\varepsilon)=U_{q,p}\rho_{q,p}(t_0)U_{q,p}^\dagger
The initial and the final density operators for the observed system can be obtained by partial trace:
\rho_q(t_0)&=\textrm{tr}_p\rho_{q,p}(t_0)\\
\rho_q(t_0+\varepsilon)&=\textrm{tr}_p\rho_{q,p}(t_0+\varepsilon),
and we have
$$\rho_q(t_0+\varepsilon)=tr_p(U_{q,p}\rho_{q,p}(t_0)U_{q,p}^\dagger).
In general, the transformation from $\rho_q(t_0)$ to $\rho_q(t_0+\varepsilon)$ is not unitary.
We perform now another simplification, again without loosing generality, by purifying the state. We can purify the state $\rho_{q,p}$ by expanding the state space from $\mathcal{S}_q\otimes \mathcal{S}_p$ to
$$\mathcal{S}:=\mathcal{S}_q\otimes \mathcal{S}_p \otimes \mathcal{S}'_q\otimes \mathcal{S}'_p,
with $\mathcal{S}_q\cong \mathcal{S}'_q$ and $\mathcal{S}_p\cong \mathcal{S}'_p$. The two extra state spaces $\mathcal{S}'_q$ and $\mathcal{S}'_p$ does not necessarily represent physical systems, but they allow us to consider $\rho_{q,p}$ as the partial trace of a pure state on $\mathcal{S}$. The composed system's evolution can be considered to be described by Schrödinger's equation on $\mathcal{S}$, although the state $\rho_{q,p}$ still needs to obey Liouville - von Neumann equation.
We denote the state space which is external to our observed system by
$$\mathcal{S}_e:=\mathcal{S}_p \otimes \mathcal{S}'_q\otimes \mathcal{S}'_p,
and the density operator describing the evolution on this space by $\rho_e$.
The conditions imposed by the observations to the system described by $\rho_q$ at $t_0$ and $t_1$ imply that $\rho_q(t_0)$ and $\rho_q(t_1)$ represent pure states:
$$\begin{array}{l}
\rho_q(t_0)=|\psi_0\rangle\langle\psi_0|\textrm{ and} \\
\rho_q(t_1)=|\psi_1\rangle \langle\psi_1|.
This imposes restrictions also on the combined system $\rho_{q,p}$. After $t_0$ the systems $\rho_q$ and $\rho_p$ become entangled, and the second observation disentangles them, and also imposes to $\rho_e$ a purity condition
$$\rho_e(t_1)=|\eta_1\rangle \langle\eta_1|,
with $\eta_1\in\mathcal{S}_e$.
Since at $t_0$ the preparation device and the observed system were being separated, the preparation device was in a state $\rho_p(t_0)$, which can be obtained by partial tracing from a pure state $|\eta_0\rangle\in \mathcal{S}_e$. Although the state vector $|\eta_1\rangle$ is uniquely determined by the observation at $t_1$, it depends on $|\eta_0\rangle$. Because we don't know the value of $|\eta_0\rangle$, to each possible $|\eta_0\rangle$, and to each possible outcomes $|\psi_0\rangle$ and $|\psi_1\rangle$ of the two measurements, will correspond a unique $|\eta_1\rangle$. In order to clarify this correspondence, we need to study some properties of linear operators acting between tensor products of vector spaces.
Let $\mathcal{V}_A$, $\mathcal{V}_B$, $\mathcal{V}_C$ and $\mathcal{V}_D$ be four vector spaces over a field $\mathbb{K}$, $\mathbb{K}=\mathbb{R}$ or $\mathbb{K}=\mathbb{C}$, and let
$$T:\mathcal{V}_A\otimes\mathcal{V}_B\to\mathcal{V}_C\otimes\mathcal{V}_D
be a $\mathbb{K}$-linear morphism of vector spaces. We are interested in identifying the possible separable vectors $|A\rangle \otimes|B\rangle \in\mathcal{V}_A\otimes\mathcal{V}_B$ and $|C\rangle \otimes|D\rangle \in\mathcal{V}_C\otimes\mathcal{V}_D$ such that:
$$T(|A\rangle \otimes|B\rangle )=|C\rangle \otimes|D\rangle .
Proposition 1.
Let us consider $|A\rangle$ and $|C\rangle$ fixed. The set of vectors $|B\rangle \in\mathcal{V}_B$, and the set of vectors $|D\rangle \in\mathcal{V}_D$, satisfying the equation above, form vector subspaces $\mathcal{V}_B^{AC}\leq\mathcal{V}_B$, respectively $\mathcal{V}_D^{AC}\leq\mathcal{V}_D$.
If $|B'\rangle$ and $|B''\rangle$ are such that $T(|A\rangle \otimes|B'\rangle )=|C\rangle \otimes|D'\rangle$ and $T(|A\rangle \otimes|B''\rangle )=|C\rangle \otimes|D''\rangle$ for some $|D'\rangle$ and $|D''\rangle \in\mathcal{S}_D$, then for any $z',z''\in\mathbb{C}$,
$$T(|A\rangle \otimes(z'|B'\rangle +z''|B''\rangle ))=z'T(|A\rangle \otimes|B'\rangle )+z''T(|A\rangle \otimes|B''\rangle )=\\
z'|C\rangle \otimes|D'\rangle +z''|C\rangle \otimes|D''\rangle =|C\rangle \otimes(z'|D'\rangle +z''|D''\rangle ),
therefore the solutions $|B\rangle \in\mathcal{V}_B$ form a vector subspace $\mathcal{V}_B^{AC}\leq\mathcal{V}_B$. Consequently, the solutions $|D\rangle \in\mathcal{V}_D$ form a vector subspace $\mathcal{V}_D^{AC}=T(\mathcal{V}_B^{AC})\leq\mathcal{V}_B$.
Remark 1.
Since we have $\mathcal{V}_C\otimes\mathcal{V}_D\cong\mathcal{V}_D\otimes\mathcal{V}_C$, it follows from Proposition 1 that a similar result holds for $|A\rangle$ and $|D\rangle$ fixed.
If the space $\mathcal{V}_C$ has a scalar product $\langle \;|\;\rangle$, the linear operator
$$T_B^{AC}:=T|_{\mathcal{V}_B^{AC}}
is given, for $|C\rangle \neq 0$, by
$$T_B^{AC}(|B\rangle )=\frac{\textrm{tr}_C(T(|A\rangle \otimes|B\rangle )\otimes\langle C|)}{\langle C|C\rangle}.
To remove the $|C\rangle$ part from $|C\rangle \otimes|D\rangle$, we tensor $|C\rangle \otimes|D\rangle$ with $\langle C|\in\mathcal{V}_C^*$, partial trace over $|C\rangle \langle C|$, and then divide by $||C\rangle |^2$.
If $T$ defined above is isomorphism, then $T_B^{AC}$ is isomorphism onto its image.
We can now apply the previous results to a unitary operator $U$ acting on our space $\mathcal{S}_q\otimes\mathcal{S}_e$:
$$U:\mathcal{S}_q\otimes\mathcal{S}_e\to\mathcal{S}_q\otimes\mathcal{S}_e,
and to the equation
$$U(|\psi_0\rangle\otimes|\eta_0\rangle)=|\psi_1\rangle \otimes|\eta_1\rangle,
obtaining the following corollaries.
Corollary 1.
Let us consider $|\psi_0\rangle$ and $|\psi_1\rangle$ fixed. The set of vectors $|\eta_0\rangle\in\mathcal{S}_e$, and the set of vectors $|\eta_1\rangle \in\mathcal{S}_e$, satisfying the equation above, form isomorphic vector subspaces $\mathcal{S}_{e0}^{\psi_0\psi_1}\leq\mathcal{S}_e$, respectively $\mathcal{S}_{e1}^{\psi_0\psi_1}\leq\mathcal{S}_e$.
Follows immediately from Proposition 1. and Remark 2.
A measurement at $t_1$, although determines the observed state to be in $|\psi_1\rangle$, it does not necessarily determine completely the state $|\eta_1\rangle$ of the preparation device.
Let us consider $|\psi_0\rangle$ and $|\eta_1\rangle$ fixed. The set of vectors $|\eta_0\rangle\in\mathcal{S}_e$, and the set of vectors $|\psi_1\rangle \in\mathcal{S}_q$, satisfying the equation from Remark 2, form isomorphic vector subspaces $\mathcal{S}_{e0}^{\psi_0\eta_1}\leq\mathcal{S}_e$, respectively $\mathcal{S}_{q1}^{\psi_0\eta_1}\leq\mathcal{S}_q$.
Follows from the Remark 1.
We denote the isomorphism obtained by restricting the unitary operator $U$ to $\mathcal{S}_{e0}^{\psi_0\eta_1}$ by $K_{\psi_0\eta_1}$.
From Corrolary 2 we obtain:
Theorem.
The set of all states $|\psi_1\rangle \in\mathcal{S}_q$ that can appear in the right side of the equation in Remark 2, for a fixed $|\psi_0\rangle\in\mathcal{S}_q$ is given by the following union of subspaces:
$$\mathcal{S}_{q1}^{\psi_0}:=\bigcup_{|\eta_1\rangle \in\mathcal{S}_e}\mathcal{S}_{q1}^{\psi_0\eta_1},
obtained by varying the state vector $|\eta_0\rangle$ in the set
$$\mathcal{S}_{e0}^{\psi_0}:=\bigcup_{|\psi_1\rangle \in\mathcal{S}_q}\mathcal{S}_{e0}^{\psi_0\psi_1}.
A good preparation must satisfies the condition
$$\mathcal{S}_{q1}^{\psi_0} \geq \{|\psi_1\rangle \in\mathcal{S}_q|\langle\psi_1|U(t_1,t_0)|\psi_0\rangle\neq 0\},
where $U(t_1,t_0)$ is the evolution operator of the observed system, if the system is undisturbed.
We recall that the state space $\mathcal{S}_e$ is an extension of a $\mathcal{S}_q$, made for working with purified states, but this is not a problem, since we can always recover the density operators of the subsystems by partial tracing.
The mechanism proposed here is represented in the figure 3. The preparation should consist in an interaction with the property that any possible outcome $|\psi_1\rangle$ of the second measurement can be fitted by an appropriate choice of the initial conditions for the preparation device, represented by the state vector $|\eta_0\rangle$.
Figure 3 Each possible outcome $|\psi_1^i\rangle$ can be obtained by choosing the appropriate states $|\eta_0^i\rangle$ representing the preparation device.
The smooth projection mechanism
Because the first measurement can find the observed system in the state $|\psi_0\rangle$, while the second one in $|\psi_1\rangle \neq U(t_1,t_0)|\psi_0\rangle$, it is easy to understand why it seemed that the state vector suffers a discontinuous jump, somewhere between $t_0$ and $t_1$. But we can now explain the wave function collapse as taking place smoothly, restoring the continuity in its evolution.
In order to do this, we had to go to the level of a larger system, composed by the observed system and the preparation device. At that level, the unitary evolution has been restored, and we have seen that the observed system (although its evolution may no longer be unitary, being entangled with the preparation device) can undergo a "smooth collapse".
The price to be paid was the acceptance that the observed system acts, somehow, anticipating the set of possible eigenstates. This feature may seem acausal, but it is present also in the standard Quantum Mechanics, as we have learned from the "delayed-choice experiments". In this article, the collapse was only pushed to the "beginning of times", and the initial conditions remained at the time $t_1$, being thus "delayed initial conditions".
Each measurement specifies the initial conditions of a system. When a system is measured a second time, the initial conditions need to be restated. To be restated without contradicting the previously observed initial conditions, they should be lost somehow. I hypothesized here that they are lost because of the interaction with the preparation device, which, although determines the previous set of conditions, transfers from its own indeterminacy of initial conditions to the observed system. Any interaction of a system with another system which has some freedom in the choice of its initial conditions, will make the former system loose its specification of the initial conditions. The observation only shows what the state was, and not what it will be at the next measurement.
Our mechanism allows us to see the projection, usually being associated to the wave function collapse, as taking place continuously, smoothly, and not discontinuously. The projector operator is not present explicitly in the evolution equation, but it is "embedded" in a set of operators parameterized by $|\eta_0\rangle$ - it can be reconstructed, for each pair $(|\psi_0\rangle,|\psi_1\rangle )$, by choosing an appropriate $|\eta_0\rangle\in\mathcal{S}_e$.
Smoothness
This article provides a scenario of how the wave function collapse can take place without discontinuities, in a smooth way. We can reconstruct the Quantum Mechanics into a Smooth version, but we have to remember that this is not the only place where discontinuities occur. For example, the eigenstates of the position are distributions, and the eigenstates of the momentum have infinite norm. If we consider the state space as being a Hilbert space, then we have to accept such problems. Yet, we can avoid this kind of problems by renouncing at the completeness - the idea that the state space should contain limits for any Cauchy sequence. We can instead use a rigged Hilbert space, $\mathcal{S}\subset \mathcal{H}\cong\mathcal{H}^*\subseteq\mathcal{S}^*$. The state vectors will then be elements of a space $\mathcal{S}$ of smooth functions of finite norms, but the (ideal) eigenstates of various operators will belong to $\mathcal{H}$.
The evolution equations are deterministic, and since we have eliminated the discontinuities, the only source of randomness is in the initial conditions. Therefore, both the Born rule and the Heisenberg relations have to be reinterpreted. The Born rule doesn't expresses the probabilities of collapse, but those of the initial conditions to lead to each outcome.
We can derive the original Heisenberg relations [WZ83] by multiplying the relations $\Delta\omega\Delta t\geq 2\pi$ and $\Delta k_x\Delta x\geq 2\pi$, from the Fourier analysis, with the reduced Plank constant $\hbar$.
To obtain similar Heisenberg relations for other pairs of conjugated operators, we do the same for the corresponding eigenbases.
These relations refer to how large the support of a state vector can be, when expressed in two different bases, and have nothing intrinsically probabilistic built in.
For example, the relation $\Delta k_x\Delta x\geq 2\pi$ shows that if the wave packet is too located in space, then in the momenta space it will be more spread.
We can obtain also the Heisenberg's relations from the commutation relations of the operators.
A version of Heisenberg's relations, which is used frequently, is $\sigma(p_x)\sigma(x)\geq \frac 1 2\hbar$, expressed in terms of the standard deviation, defined for an operator $A$ by $\sigma(A):=\sqrt{\langle A^2\rangle-\langle A\rangle^2}$. Again, the probabilities have not yet entered into the play, because the standard deviations, in this case, refer to the components of the wave packet, expressed in two conjugate bases.
It is only when the state vector is disturbed by a preparation, and we apply the Born rules in relation to an eigenbasis of an observable, when Heisenberg's relations become the uncertainty relations. It follows that the probabilistic meaning of the Heisenberg's relations also reflects our ignorance of the initial conditions.
The observers don't have access to the full set of initial conditions. The observations allows them collect only a set (which we will name registry) of partial initial conditions. Therefore, although the evolution is deterministic, they perceive the time evolution as being indeterministic.
Causality and delayed initial conditions
The mechanism in Smooth Quantum Mechanics resumes to fixing the initial conditions at a moment of time, even for events prior to that moment. In fact, from mathematical point of view, the same solution of a PDE can be obtained from appropriate initial conditions imposed at any instant of time.
The main motivation for choosing the solution of delayed initial conditions resides in the avoidance of discontinuities. The discontinuities are source of problems. They imply that there are two sets of physical laws at quantum level. The discontinuities have never been observed directly, only the incompatibility between the outcomes of the measurements. They require additional explanations for the conservation laws, which are respected always, although only the smooth evolution leads to them, as they are obtained from commutation with the Hamiltonian (or by Noether's theorem). The price for avoiding the discontinuous collapse is to allow the initial data to be specified with a delay.
But we should clarify whether or not the mechanism of delayed initial conditions breaks down the causality. As already mentioned, each new condition should be chosen such that it is compatible with the already chosen conditions. It is even possible to choose two or more conditions at positions and moments which can be related only with signals traveling with velocity greater than that of the light. In this case, if we account for relativity, we can add the two conditions to the registry simultaneously, taking care not to violate the compatibility. In a deterministic world, with the initial data completely specified, there is no room for phenomena which contradict the initial data. Similarly, in the Smooth Quantum Mechanics, all the initial data is required to be compatible with the initial data already acquired. A similar apparent acausality, manifested by the anticipation of future initial conditions, is also present in experiments with photons having negative group velocity [GSBKB06]. Consequently, the causal loops and breaking of causality are avoided.
What remains to be done
This article only shows that it is possible to have a smooth, instead of a discontinuous, wave function collapse, and shows that it is possible a smooth reconstruction of Quantum Mechanics. Not any interaction is able to provide the freedom in initial conditions required to solving this problem. Perhaps, this is why not any interaction is a measurement, but this point needs to be developed better. Ideally, we would have a precise mathematical description of a measurement, and a theorem showing that from this description, we obtain precisely the required range of outcomes at a second observation. Having a good definition of the measurement apparatus will allow us to predict, for example, which interactions qualify as measurements. Maybe, for this understanding, we will have to wait until more challenging parts of the Quantum Mechanics - the reconstruction of the classical world from the quantum world, and the explanation of why a measurement can obtain only eigenstates of the observable as outcomes - will receive better explanations. Another important progress would be a deduction of the Born rule. At the current moment, it seems that this rule is independent on the Smooth Quantum Mechanics' principles, but it would be desirable to have at least a good definition of a measurement which will lead easier to a smooth version of the Projection Postulate, including the Born rule.
Relations with other interpretations of Quantum Mechanics
After about eight decades of progresses in Quantum Mechanics, the discussions between Einstein and Bohr remain of actuality.
Although their views seemed incompatible one another, the Smooth Quantum Mechanics presented here is friendly with both of them. I don't say that, had they living today, they would say that they had in mind this solution, but I hope that this is at least a small step towards a reconciliation between their viewpoints. In a way, Bohr was right to say that "a phenomenon does not exist, until is observed" [Bohr28, WZ83], and Einstein was right to hope for a better, more complete, explanation of the quantum phenomena.
Perhaps, Schrödinger's ideas are most compatible with the SQM, since he disliked the discontinuous collapse, and believed in the physical reality of the wave functions. For example, he took the charge density in the electron's wave function literally, not as a probability distribution, and, according to SQM, he may be right.
The electron is the electron's wave function, since it is not a point, it is a wave, having different "shapes", depending on the measured observables.
The determinism is regained, since the evolution is again deterministic. The efforts of de Broglie, Vigier, culminating with David Bohm's causal or ontological interpretation of Quantum Mechanics [Bohm52, BH93], are theories whose purpose is to restore the determinism, the causality, and the reality and independence of the world. The price, as we now know, was the necessity to admit the nonlocality [Bel64, CHSH69, CS78, ADR82, Asp99]. SQM provides a deterministic theory without extra "hidden variables", rather, it is based on undetermined variables, or undetermined initial data. Here, "to determine" has a passive meaning - "to measure/observe", and an active one - "to choose". The initial data is determined by measurements, but we can choose what to measure. We can look at the indeterministic QM as applying to open systems only, whose description can be completed to a deterministic image by accounting for the parameters "hidden" in the environment.
The indeterministic view is not completely lost, since what the observer has is the registry, which is never a complete set of initial data. Each new measurement can bring new information, and the registry can be extended in different ways. We can interpret this in two ways. The first way is that the past is not established, and it is progressively created by each new choice of the observables, and, consequently, by each new outcome of the measurement. The second way to see the things is that all possible worlds exist, like a sheaf, and when we choose the observable we reduce the sheaf of worlds compatible with our registry. Each extension of the registry reduces this sheaf. Therefore, SQM is compatible with the Many Worlds Interpretation, with the amendment that each world is deterministic, and the only split is in the observer's registry, which can be completed in many ways. We can call this version of MWI the Many Registry Interpretation.
Perhaps one reason in the acceptance of a fundamentally indeterministic behavior in Quantum Mechanics was the belief that this is the only way to allow the existence of free-will [CK06, CK08]. The Smooth Quantum Mechanics offers an alternative, a deterministic view, which is still compatible with the free-will, at the same extent as the standard QM. We have the same freedom in choosing what observable to measure, influencing by this the initial conditions [tH07], but in a smooth and deterministic manner.
[SMKYKS00]
R.Y.S.K. Scully, Marlan O.; Yoon-Ho Kim, Y. Shih, Physical Review Letters (84:
1-5) (2000).
arxiv:quant-ph/9903047
[Whe77]
J. Wheeler, in Quantum Mechanics a Half Century Later (1977), pp. 1--18
J.A. Wheeler, in In A.R. Marlow, ed., Mathematical Foundations of Quantum Theory (1978), p. 30
[WZ83]
J.A. Wheeler, W.H. Zurek.
\newblock Quantum theory and measurement (1983)
[Wei31]
K.F. Weisz\"acker, Zeit. F. Phys. (70), 114 (1931)
[Sch49]
P.A.S. Ed., Albert Einstein : Philosopher-Scientist. The library of living philosophers, vol VII. (Cambridge University Press, 1949)
[EPR35]
A. Einstein, B. Podolsky, N. Rosen.
\newblock Can quantum-mechanical description of physical reality be considered
complete?, physical review 47:777--780 (1935)
[Bohm51]
D. Bohm, Quantum Th. pp. 611--623 (1951)
[GSBKB06]
G.M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski, R.W. Boyd, Science
(312), 895 (2006)
[Bohr28]
N. Bohr, Naturwissenschaften \textbf{16}, 245 (1928).
\newblock \doi{10.1007/BF01504968}
D. Bohm, Phys. Rev (85) (1952)
[BH93]
D. Bohm, B. Hiley.
\newblock The undivided universe: an ontological interpretation of quantum
mechanics. routledge and kegan (1993)
[Bel64]
J.S. Bell, (1964)
[CHSH69]
J.F. Clauser, M.A. Horne, A. Shimony, R.A. Holt, Physical Review Letters (23)
[CS78]
J.F. Clauser, A. Shimony, in Reports in the Progress of Physics (1978)
[ADR82]
A. Aspect, J. Dalibard, G. Roger, Physical Review Letters (49) (1982)
[Asp99]
A.B. Aspect.
\newblock Bell's inequality test: More ideal than ever. nature 398 (1999)
[CK06]
J. Conway, S. Kochen, Found. Phys. (36), 1441 (2006)
J. Conway, S. Kochen,
[tH07]
G. 't Hooft.
The free-will postulate in quantum mechanics (2007).
Labels: Smooth Quantum Mechanics
Smooth Quantum Mechanics: 4. The Video
Here is a five minutes video explaining the main idea of the Smooth Quantum Mechanics.
What is the collapse of the wave function? Is it necessary? The Quantum Mechanics can be built without making use of discontinuities in the time evolution. The appearance of the collapse is due to the entanglement between the preparation device and the observed system, in combination with the delayed choice initial conditions.
Smooth Quantum Mechanics: 3. Registry and Evolving Block Universe
The time evolution appears to humans as a flow, subject to our free-will. The past appears to be frozen forever, and the future seems open to various possibilities. This post compares this view, of Evolving Block Universe, due to Professor George Ellis, with the registry approach of Smooth Quantum Mechanics.
Registry and Evolving Block Universe
The block view and the time evolution view are not as incompatible as they may look at a first sight. We can recover the time evolution by watching the entropy distribution between the events of the block world, and the causal co-relations between them. I would like to compare the registry time evolution with the Evolving Block Universe of George Ellis. Professor Ellis proposes an evolving block universe, perhaps the most credible proposed so far. He explains that the quantum phenomena (in the standard indeterministic interpretation of QM) should have gravitational effects. Consequently, they must change the spacetime. I agree with this argument. Further, he details a theory in which the time flows, evolves, in a sort of presentist way, and the past, which already happened, is "archived" in a block universe. The block universe increases with time, as new "presents" adds to it. The future is not decided yet, and as it happens, it becomes present, and then it is archived. This view is well elaborated, and reflects well our feelings of time flow, free-will, carved in stone past, and open future. On the other hand, I do not agree with Professor Ellis that QM proves the indeterminism. Even so, assuming the indeterminism valid, this doesn't eliminate the possibility of the standard block view.
Perhaps the most important difficulty of such an Evolving Block Universe is the possibility, offered by QM, of deciding the past events chronologically after they took place. This implies that we have to wait to archive the passed times. Moreover, it is possible to never be able to determine the past completely. Consider Wheeler's delayed choice experiment, with the photon emitted by a distant star. The observer watching the star will decide whether to measure the "both ways", or the "which way". Her decision affects the past history of the observed photon, hence of the observed star. Of course, it is unlikely that she affected the star's state in a significant way, but she affected it at least in a small way. Until the observation, the photon, hence the star (by entanglement), was in an undefined state. Assume now that the photon is never observed, and escapes far from any planet and any possible observer. The Universe will remain in an undetermined state. So, we cannot say that the past block will be ever created. On the other hand, my proposal of a "registry" of incomplete initial data which increases with each observation, relying on Smooth Quantum Mechanics, allows the possibility that the state of the Universe remains undetermined. Professor's Ellis idea of foliating the spacetime so that the spacelike surfaces contains the wavefunction collapses may be unreachable, because the entanglement makes the collapses impossible to be ordered temporally. I am afraid that the entanglement can be complicated enough. The measurements of the spins of the two electrons in the EPR-B experiment can be in any spacetime relation. We cannot consider that the wavefunction collapse takes place necessarily along such preferred spacelike surfaces, which are compatible with a spacetime foliation. It is easy to see that, if we associate spacelike surfaces to the collapse, it is possible that these intersect in complicated ways. Moreover, collapse can take place also between events that cannot belong to the same spacelike surface, being for example one in the other's future.
The standard BU attempts to express the temporal structures in terms of timeless structures. We can consider it, in a way, as a research program of explaining the time itself in terms of timeless structures. But, by adhering to a presentist view, and by reducing the BU functionality to a purely archiving role, there is the danger of explaining the time by appealing to time in a circular way: the EBU includes the passed time in the archived BU, but the evolution happens in a metatime. Another interesting feature the BU has is that it contains all the physical fields in its description. By giving a special role to the present, we introduce a feature which has no correspondent in the matter fields. The BU accounts for the physical fields, but it cannot include an intrinsic present, and maybe doesn't even need. Yet, if it would need to mark the present, a "BU with a bookmark" would solve the problem.
The registry view is compatible with both time evolution, and with the standard block universe view. And it shares with the EBU picture the compatibility with our feelings of time flow, free-will, open future, but not the carved in stone past.
Smooth Quantum Mechanics: 2. Registry and Time Evolution
In the Smooth Quantum Mechanics, the evolution is deterministic, but the initial data is not determined. The registry is a collection of partial initial data, which is in general incomplete. This leads in particular to a version of the Many Worlds Interpretation: the Many Registries Interpretation.
Registry and Time Evolution
In the Smooth Quantum Mechanics, the evolution is, for the complete system, deterministic and unitary. The randomness appears because of the incomplete knowledge of the initial conditions. Not only that we don't know these initial conditions, but they are even not defined, until we perform the measurement. This is because of Bell's theorem. We can choose what observable to measure, and each observable limits the possible outcomes in a different way. We can have two observables so that it is clear that their outcomes are relatively incompatible. When we choose one of them, we chose what already happened. We determine the past in the active sense of the word "to determine".
Each observation we make increases the constrains of the solution of the evolution equation. The set of (delayed) initial conditions known at a time is named "the registry". This registry increases as new measurements are related causally to the registry. For the observer, the time evolution correlates with the registry expansion. Some measurements are independent, and others are correlated, since they measure entangled particles or degrees of freedom. The registry is a network of such events, and the way it expands is correlated with the time arrow. The physical laws being time symmetric at a fundamental level, the relations between the measurements are probably correlated with the thermodynamic time arrow. Therefore, the registry can be viewed in a timeless way, providing a block world (or block universe, or BU) view. On the other hand, it can be viewed as being correlated with the time's arrow, providing an evolving world view.
Many Registries Interpretation of Quantum Mechanics
The way the registry expands by adding new observations is not determined by the present state of the registry (except when the registry is complete, providing a full description of initial data). This means that we can expand the registry in many ways. These multiple possibilities remind us the Many Worlds Interpretation of Quantum Mechanics, in which the world splits with each new measurement, according to each outcome. There is a difference, in that the split in the MWI is due to the indeterministic character of the wavefunction collapse (although the "total" wavefunction evolves deterministically). In the Many Registries Interpretation, each world is deterministic, but the observer has not identified/chosen yet her world, so she perceived the evolution as indeterministic. By adding new observations to the registry, she can increase the information about the world. Thus, she selects the world, and she even seems to have a small possibility of choosing the world, by choosing the observable.
By adhering to Smooth Quantum Mechanics, we can keep the idea of MWI, in the form of Many Registries Interpretation. The idea of "registry" reconciles the indeterminism perceived by the observer, with the fundamental determinism of the unitary evolution.
Even if we don't know if we will ever be able to provide experimental support for one or another version of the MWI, I think that the MWI has at least a great pedagogical value. Also, it allows a better reconciliation of the block world with the apparent openness of the future.
Smooth Quantum Mechanics: 1. The Smooth Particle
Smooth Quantum Mechanics provides a way to avoid the discontinuity in the wavefunction collapse. This post presents the wave-like nature of the particles in Smooth QM, as compared to the standard QM. The probabilities also have a different character, being rooted in the initial conditions, instead of discontinuities.
In Smooth Quantum Mechanics, there are only (entangled) waves.
In standard QM, the particle behavior is manifest when the observable is a position operator acting on the Hilbert space. In this case, the system is found in an eigenstate of the position operator, which is a Dirac distribution (a generalized function which is zero everywhere, except in one point, where is infinite, such that its integral is 1). The wave and particles are, in standard QM, all vectors in the Hilbert space, being therefore wave functions or distributions. When expressed as wavefunction (in a basis of positions), the "pure waves" are eigenstates of momenta, and the "pure Dirac distributions" are eigenstates of positions - these are extreme situations, in which infinities occur.
One interpretation of the waves in QM is that they underlie probabilities of finding the particles at a given point in space. Of course, this is true, grace to the Born rule, but this does not necessarily mean that the point particles are the fundamental ones, and the wave represents only probabilities. Yet, many like to think at particles as being fundamentally point-like, therefore the waves need to be interpreted as underlying probabilities.
In Smooth QM, it is considered that the only real physical states of the system are smooth: it is employed a space of smooth wavefunctions of finite norm. By completing the space, we obtain a rigged Hilbert space, which contains state vectors that are abstract, being distributions or having infinite norm. By observing the position, we get only a smooth wave concentrated around that position, but not a Dirac distribution. The position is never determined to be in a point, but in an open subset of the space. I think that the two extremes: eigenstates of pure momenta and pure positions, are non-physical abstractions, because they bring in infinities, but they are useful to express and explain the wavefunction duality. In Smooth QM, by not allowing the distributions, by eliminating the infinities, the unity between the two complementary aspects is more manifest and more physical. All particles are waves, in various shapes, depending on the observable. Of course, the Hilbert space is very useful, and it is easy to solve the equations in this space, and many operations become simpler. But in Smooth QM it is considered abstract. The wave is physical, not a probability wave.
Therefore, we return to the old idea of particles as wave packets. One main problem of this view was the dispersion of the wave packet. In Smooth QM, the laws are the same, so the dispersion is not eliminated, but the mechanism is such that the dispersion is no longer a problem. A measurement of position, finding the wave-like particle localized very well, will imply dispersion both in the future, and in the past. The waves converge, the wave packet is "con-persed" until the small sized packet is obtained and determined by measurement, and then it disperses again. The "delayed initial conditions" mechanism allows this.
The probabilistic behavior occurs only when we measure an observable which does not contain the observed state among its eigenstates. In this case, the Born rule expresses the probabilistic character, but the probability in Smooth QM resides only in the undetermined initial conditions. Here "determined" has two meanings, active and passive: to cause, and to measure. The probabilistic behavior occurs when a collapse occurs, but in this approach the collapse is smooth and deterministic, and the probabilities reside in the initial conditions. Even the Heisenberg relations, which are often thought as representing irreducible probabilities, are not necessarily such. They can be obtained from Fourier analysis, and apply for deterministic waves. The probabilistic character occurs because of measurements, which entails the (smooth) collapse, which brings the probabilities of the unknown initial conditions.
Although I said that the waves are physical, and real, we should not forget that they are entangled. For a particle, the wave can be a field depending on space and position, but for more particles, we need to consider the tensor products, therefore the entanglement.
Posted by Cristi Stoica at 10:56 PM
The Counterintuitive Time: 5. Quantum Time
The counterintuitive nature of time in Physics series continues with Quantum Mechanics, with entanglement and delayed choice experiments. It is presented the Smooth Quantum Mechanics, which eliminates the discontinuity from the wavefunction collapse. It happens to be deterministic, but the compatibility with free-will is maintained.
Nonrelativistic Quantum Mechanics describes a system by a vector, named state vector, from a complex Hilbert space (a special type of complex vector space, endowed with a special type of scalar product). To the classical quantities, we associate selfadjoint operators on the Hilbert space. The space differs from the time, because there are position operators, while the time is only a parameter.
Schrödinger proposed an equation, describing the evolution of the state of a system. Schrödinger's equation is of PDE type, and it is deterministic, linear, even unitary (it preserves the scalar product). What we can observe or measure is an operator, representing the observable we want to measure. What we can get as outcome, is that the state vector of the system is one of the observable's eigenvectors (special vectors associated to each operator). This means that we can never know what the system's state is, without disturbing it, because there are few chances that the system is already in an eigenstate of the observable.
In the standard interpretation, the system jumps into one of the eigenstates of the observable. We cannot know before in which, but we can know the probability for each possible outcome, due to Born's rule. This introduces the indeterminism at the very fundamental level of reality. The time gains a strange feature, because it appears that, at any moment, a system can jump in a state without an apparent cause. The Classical Mechanics paradigm identifying the causality with the deterministic evolution lasted for centuries. QM introduced the possibility that a system jump out of the blue, and opened a totally different perspective. To resolve some problems of QM, Hugh Everett III proposed an interpretation of QM which states that each possible jump takes in fact place, but the world splits in many worlds, each of them containing one of the possible jumps. In this interpretation, time itself looks like it is branching, or forking, although the observers cannot check the existence of the other alternative histories. Despites the fact that for each observer, "prisoner" of one of these worlds, the wavefunction collapse and other strange quantum phenomena remain unexplained as before, this interpretation offers a intuitive and unitary view of what happens.
Some of the founders of QM, Einstein, de Broglie, Schrödinger, felt that accepting the indeterminism means to give up the search for a better explanation. Nowadays, when the indeterministic view in QM is well established, they are sometimes presented like conservators, with little understanding of quantum phenomena. This is unfair, because not only they co-initiated the quantum revolution, together with Bohr, Born and Heisenberg, but they also expressed the problems which this new born theory encountered, this leading to a refinement of the theory and its interpretations. Schrödinger explained the idea of entanglement, which springs from the very fundamental principles of Quantum Mechanics. Einstein, Podolsky, and Rosen, proposed an experiment which showed a paradoxical behavior of quantum mechanics, which is in fact the entanglement between two particles that previously interacted. This brings a weird aspect of time: they interacted in the past, and now, by measuring one of them, we can limit the possible outcomes of a measurement performed to the other one. It appears that the wavefunction has a nonlocal character over space and time.
One strange quantum effect is visible by the "delayed choice experiments", made popular by Wheeler. Wheeler provides the example of a photon emitted by a very distant star. He considers the case when between us and that star there is a galaxy, which bend the light ray, according to General Relativity. According to QM, among the possible experiments we can make with the incoming photon, there are two mutually exclusive. First, we can observe whether it passes through the left, or through the right of that galaxy - the "which way" measurement. The second possibility is to put the two possible ways to interfere one another, like the photon was traveled "both ways". The problem is that we can make our choice now, long time after the photon was emitted by the distant star, and long time after it was bent by that galaxy. We can choose now what kind of behavior had the photon thousands of years ago. This is really something that bends our intuition on time very much. We tend to believe that the past determines, or at least influences the future, but future influencing the past?
It is usually believed that the wavefunction, when measured, suffers a collapse. The corresponding state vector becomes suddenly projected on one of the observable's eigenstates. This is a little strange, because it entails a discontinuity in evolution, which we never observed. This discontinuity makes more difficult the preservation of conserved quantities, because usually the conservation laws are effects of the unitary evolution, but a discontinuous jump may break them down. Yet, we haven't observed such breaking of the conservation laws, nor we had observed other direct evidence of the jump, except our knowledge that we prepared the system to be in one state, and we detect it in another state. In the Smooth Quantum Mechanics eprint, I show that we can avoid the discontinuity of the wavefunction collapse. I use the entanglement between the observed system, and the measurement device that performed the previous measurement (the preparation device), and the possibility of choosing with a delay the initial conditions. What appears to be a jump, is described in a continuous, even smooth way (which is even unitary at a higher level). The past interaction with the preparation device happens in such a way, that it anticipates the outcome of the measurement. This interaction takes place during a finite time, and changes smoothly the state, such that, when it is measured, to be an eigenstate of the observable. I use a mechanism similar to the delayed choice experiment, but which, because of the smoothness, extends indefinitely in the past.
Because the smooth QM provides a smooth description of what was believed to be a discontinuous collapse, it appears that Einstein, de Broglie, and Schrödinger weren't that wrong. The determinism was also brought back by Bohm, by using nonlocal hidden variables. In the smooth QM, the hidden variables are replaced naturally by the yet to be determined initial conditions. The nonlocality remains in all versions, but the determinism becomes possible just by unitary evolution (Schrödinger's equation being replaced with the von Neumann's, because we deal with entangled states). So, we can say that both sides in the Einstein-Bohr debate were simultaneously right, at an unexpected degree.
If the standard QM allows the free-will, so does the smooth version, because the freedom of choosing the observable is exactly the same. The smooth version is deterministic, but the initial conditions are not determined yet completely, and each new experiment adds new information about them. This is why they can be named "delayed initial conditions".
Labels: Time
Smooth Quantum Mechanics: 3. Registry and Evolving...
Smooth Quantum Mechanics: 2. Registry and Time Evo...