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That is what an interpretation of the theory would provide: a proper
account of what the world is like according to quantum mechanics,
intrinsically and from the bottom up. The problems with giving an
interpretation (not just a comforting, homey sort of interpretation,
i.e., not just an interpretation according to which the world
isn’t too different from the familiar world of common sense, but
any interpretation at all) are dealt with in other sections of this
encyclopedia. Here, we are concerned only with the mathematical heart
of the theory, the theory in its capacity as a mathematical machine,
and -- whatever is true of the rest of it -- *this* part of the
theory makes exquisitely good sense.

- 1. Terminology
- 2. Mathematics
- 3. Quantum Mechanics
- 4. Structures on Hilbert Space
- Bibliography
- Other Internet Resources
- Related Entries

A **physical quantity** is a mutually exclusive and jointly
exhaustive family of physical properties (for those who know this way
of talking, it is a family of properties with the structure of the
cells in a partition). Knowing what kinds of values a quantity takes
can tell us a great deal about the relations among the properties of
which it is composed. The values of a bivalent quantity, for
instance, form a set with two members; the values of a real-valued
quantity form a set with the structure of the real numbers. This is
a special case of something we will see again and again, *viz.*,
that knowing what kind of mathematical objects represent the elements
in some set (here, the values of a physical quantity; later, the
states that a system can assume, or the quantities pertaining to it)
tells us a very great deal (indeed, arguably, all there is to know)
about the relations among them.

In quantum mechanical contexts, the term
‘**observable**’ is used interchangeably with
‘physical quantity’, and should be treated as a technical
term with the same meaning. It is no accident that the early
developers of the theory chose the term, but the choice was made for
reasons that are not, nowadays, generally accepted. The
**state-space** of a system is the space formed by the set of its
possible
states,^{[2]}
i.e., the physically possible ways of combining the
values of quantities that characterize it internally. In classical
theories, a set of quantities which forms a supervenience basis for
the rest is typically designated as ‘basic’ or
‘fundamental’, and, since any mathematically possible way
of combining their values is a physical possibility, the state-space
can be obtained by simply taking these as
coordinates.^{[3]}
So, for instance, the state-space of a classical mechanical system
composed of n particles, obtained by specifying the values of 6n
real-valued quantities - three components of position, and three of
momentum for each particle in the system - is a 6n-dimensional
coordinate space. Each possible state of such a system corresponds to
a point in the space, and each point in the space corresponds to a
possible state of such a system. The situation is a little different
in quantum mechanics, where there are mathematically describable ways
of combining the values of the quantities that don’t represent
physically possible states. As we will see, the state-spaces of
quantum mechanics are special kinds of vector spaces, known as
Hilbert spaces, and they have more internal structure than their
classical counterparts.

A **structure** is a set of elements on which certain operations
and relations are defined, a **mathematical structure** is just a
structure in which the elements are mathematical objects (numbers,
sets, vectors) and the operations mathematical ones, and a
**model** is a mathematical structure used to represent some
physically significant structure in the world.

The heart and soul of quantum mechanics is contained in the Hilbert
spaces that represent the state-spaces of quantum mechanical
systems. The internal relations among states and quantities, and
everything this entails about the ways quantum mechanical systems
behave, are all woven into the structure of these spaces, embodied in
the relations among the mathematical objects which represent
them.^{[4]}
This means that understanding what a system is like according
to quantum mechanics is inseparable from familiarity with the
internal structure of those spaces. Know your way around Hilbert
space, and become familiar with the dynamical laws that describe the
paths that vectors travel through it, and you know everything there
is to know, in the terms provided by the theory, about the systems
that it describes.

By ‘know your way around’ Hilbert space, I mean something more than possess a description or a map of it; anybody who has a quantum mechanics textbook on their shelf has that. I mean know your way around it in the way you know your way around the city in which you live. This is a practical kind of knowledge that comes in degrees and it is best acquired by learning to solve problems of the form: How do I get from A to B? Can I get there without passing through C? And what is the shortest route? Graduate students in physics spend long years gaining familiarity with the nooks and crannies of Hilbert space, locating familiar landmarks, treading its beaten paths, learning where secret passages and dead ends lie, and developing a sense of the overall lay of the land. They learn how to navigate Hilbert space in the way a cab driver learns to navigate his city.

How much of this kind of knowledge is needed to approach the philosophical problems associated with the theory? In the beginning, not very much: just the most general facts about the geometry of the landscape (which is, in any case, unlike that of most cities, beautifully organized), and the paths that (the vectors representing the states of) systems travel through them. That is what will be introduced here: first a bit of easy math, and then, in a nutshell, the theory.

Multiplying a vector |A> by

Figure 1: Vector Addition

In a real vector space, the (inner or dot) product of a pair of vectors |A> and |B>, written ‘<A|B>’ is a scalar equal to the product of their lengths (or ‘norms’) times the cosine of the angle, , between them:

<A|B> = |A| |B| cosLet |A

|B> =For example, here is a graph which shows how |B> can be represented as the sum of the two unit vectors |Ab_{1}|A_{1}> +b_{2}|A_{2}>

Figure 2: Representing |B> by Vector Addition of Unit Vectors

Now the definition of the inner product <A|B> has to be
modified to apply to complex spaces. Let *c** be the complex
conjugate of *c*. (When *c* is a complex number of the
form *a* ± *bi*, then the complex conjugate
*c** of *c* is defined as follows:

[So, for all complex numbersa+bi]* =abi

[abi]* =a+bi

<A|B> = (a_{1}*)(b_{1}) + (a_{2}*)(b_{2})

The most general and abstract notion of an inner product, of which we've now defined two special cases, is as follows. <A|B> is an inner product on a vector space V just in case

(i) <A|A> = |A|It follows from this that^{2}, and <A|A>=0 if and only if A=0(ii) <B|A> = <A|B>*

(iii) <B|A+C> = <B|A> + <B|C>.

(i) the length of |A> is the square root of inner product of |A> with itself, i.e.,A|A| = <A|A>,and(ii) |A> and |B> are mutually perpendicular, or

orthogonal,if, and only if, <A|B> = 0.

Any collection of N mutually orthogonal vectors of length 1 in an
N-dimensional vector space constitutes an **orthonormal basis**
for that space. Let |A_{1}>, ... , |A_{N}> be such
a collection of unit vectors. Then every vector in the space can be
expressed as a sum of the form:

|B> =whereb_{1}|A_{1}> +b_{2}|A_{2}> + ... +b_{N}|A_{N}>,

Notice that:

(i) for all vectors A, B, and C in a given space,There is another way of writing vectors, namely by writing their expansion coefficients (relative to a given basis) in a column, like so:<A|B+C> = <A|B> + <A|C>(ii) for any vectors M and Q, expressed in terms of the A-basis,

|M> + |Q> = (m_{1}+q_{1})|A_{1}> + (m_{2}+q_{2})|A_{2}> + ... + (m_{N}+q_{N})|A_{N}>,and

<M|Q> =

m_{1}q_{1}+m_{2}q_{2}+ ... +m_{n}q_{n}

where

|Q> = q_{1}

q_{2}

When we are dealing with vector spaces of infinite dimension, since
we can’t write the whole column of expansion coefficients needed to
pick out a vector since it would have to be infinitely long, so
instead we write down the function (called the ‘wave
function’ for Q, usually represented
(*i*))
which has those coefficients as values. We write down, that is, the
function:

(Given any vector in, and any basis for, a vector space, we can obtain the wave-function of the vector in that basis; and given a wave-function for a vector, in a particular basis, we can construct the vector whose wave-function it is. Since it turns out that most of the important operations on vectors correspond to simple algebraic operations on their wave-functions, this is the usual way to represent state-vectors.i) =q_{i}= <Q|A_{i}>

When a pair of physical systems interact, they form a composite
system, and, in quantum mechanics as in classical mechanics, there is
a rule for constructing the state-space of a composite system from
those of its components, a rule that tells us how to obtain, from the
state-spaces, H_{A} and H_{B} for A and B, respectively, the state-space --
called the ‘tensor product’ of H_{A} and H_{B}, and written
H_{A}H_{B}
-- of the pair. There are two important things about the rule; first,
so long as H_{A} and H_{B} are Hilbert spaces,
H_{A}H_{B} will be as well, and
second, there are some facts about the way
H_{A}H_{B} relates to H_{A} and
H_{B}, that have surprising consequences for the relations between the
complex system and its parts. In particular, it turns out that the
state of a composite system is not uniquely defined by those of its
components. What this means, or at least what it appears to mean, is
that there are, according to quantum mechanics, facts about composite
systems (and not just facts about their spatial configuration) that
don’t supervene on facts about their components; it means that
there are facts about systems as wholes that don’t supervene on
facts about their parts and the way those parts are arranged in
space. The significance of this feature of the theory cannot be
overplayed; it is, in one way or another, implicated in most of its
most difficult problems.

In a little more detail: if
{*v*_{i}^{A}} is an orthonormal basis
for H_{A} and {*u*_{j}^{B}}
is an orthonormal basis for H_{B}, then the set of pairs
(*v*_{i}^{A},
*u*_{j}^{B}) is taken to form an
orthonormal basis for the tensor product space
H_{A}H_{B}. The notation
*v*_{i}^{A}*u*_{j}^{B}
is used for the pair
(*v*_{i}^{A},*u*_{j}^{B}),
and inner product on
H_{A}H_{B}
is defined
as:^{[6]}

<It is a result of this construction that although every vector in Hv_{i}^{A}u_{m}^{B}|v_{j}^{A}u_{n}^{B}> = <v_{i}^{A}|v_{j}^{A}> <u_{m}^{B}|u_{n}^{B}>

(i) any composite state defines uniquely the states of its components.(ii) if the states of A and B are pure (i.e., representable by vectors

v^{A}andu^{B}, respectively), then the state of (A+B) is pure and represented byv^{A}u^{B}, and(iii) if the state of (A+B) is pure and expressible in the form

v^{A}u^{B}, then the states of A and B are pure, but(iv) if the states of A and B are not pure, i.e., if they are mixed states (these are defined below), they do not uniquely define the state of (A+B); in particular, it may be a pure state not expressible in the form

v^{A}u^{B}.

(i) O(|A> + |B>) = O|A> + O|B>, andJust as any vector in an N-dimensional space can be represented by a column of N numbers, relative to a choice of basis for the space, any linear operator on the space can be represented in a column notation by N(ii) O(

c|A>) =c(O|A>).

where O

O = O _{11}

O_{21}O _{12}

O_{22}

O|B> =Two more definitions before we can say what Hilbert spaces are, and then we can turn to quantum mechanics. |B> is an

= O _{11}

O_{21}O _{12}

O_{22}× b_{1}b_{2}

= (O _{11}b_{1}+ O_{12}b_{2})

(O_{21}b_{1}+ O_{22}b_{2})

= (O _{11}b_{1}+ O_{12}b_{2})|A_{1}>+ (O _{21}b_{1}+ O_{22}b_{2})|A_{2}>

= |B>

A **Hilbert space**, finally, is a vector space on which an inner
product is defined, and which is complete, i.e., which is such that
any Cauchy sequence of vectors in the space converges to a vector in
the space. All finite-dimensional inner product spaces are
complete, and I will restrict myself to these. The infinite case
involves some complications that are not fruitfully entered into at
this stage.

b. *Contexts of type 2* ("Measurement
Contexts"):^{[9]}
Carrying out a "measurement" of an observable B on a system
in a state |A> has the effect of collapsing the system into a
B-eigenstate corresponding to the eigenvalue observed. This is known
as the **Collapse Postulate**. Which *particular*
B-eigenstate it collapses into is a matter of probability, and the
probabilities are given by a rule known as **Born’s Rule**:

prob(b_{i}) = |<A|B=b_{i}>|^{2}.

There are two important points to note about these two kinds of contexts:

- The distinction between contexts of type 1 and 2 remains to be made out in quantum mechanical terms; nobody has managed to say in a completely satisfactory way, in the terms provided by the theory, which contexts are measurement contexts, and
- Even if the distinction is made out, it is an open interpretive
question whether there
*are*contexts of type 2; i.e., it is an open interpretive question whether there are any contexts in which systems are governed by a dynamical rule*other*than Schrödinger’s equation.

All of the physically consequential features of the behaviors of quantum mechanical systems are consequences of mathematical properties of those relations, and the most important of them are easily summarized:

(P1) Any way of adding vectors in a Hilbert space or multiplying them by scalars will yield a vector that is also in the space. In the case that the vector is normalized, it will, from (3.1), represent a possible state of the system, and in the event that it is the sum of a pair of eigenvectors of an observable B with distinct eigenvalues, it will not itself be an eigenvector of B, but will be associated, from (3.4b), with a set of probabilities for showing one or another result in B-measurements.If we make a couple of additional interpretive assumptions, we can say more. Assume, for instance, that(P2) For any Hermitian operator on a Hilbert space, there are others, on the same space, with which it doesn’t share a full set of eigenvectors; indeed, it is easy to show that there are other such operators with which it has

noeigenvectors in common.

(4.1) Every Hermitian operator on the Hilbert space associated with a system represents a distinct observable, and (hence) every normalized vector, a distinct state, andIt follows from (P2), by (3.1), that no quantum mechanical state is an eigenstate of all observables (and indeed that there are observables which have(4.2) A system has a value for observable A if, and only if, the vector representing its state is an eigenstate of the A-operator. The value it has, in such a case, is just the eigenvalue associated with that eigenstate.

^{[11]}

There are Hermitian operators on the tensor product
H_{1}H_{2}
of a pair of Hilbert spaces H_{1} and H_{2} ... In
the event that H_{1} and H_{2} are the state spaces
of systems S1 and S2,
H_{1}H_{2} is the state-space
of the complex system (S1+S2). It follows from this by (4.1)
that there are observables pertaining to (S1+S2) whose values are not
determined by the values of observables pertaining to the two
individually.

These are all straightforward consequences of taking vectors and operators in Hilbert space to represent, respectively, states and observables, and applying Born’s Rule (and later (4.1) and (4.2)), to give empirical meaning to state assignments. That much is perfectly well understood; the real difficulty in understanding quantum mechanics lies in coming to grips with their implications -- physical, metaphysical, and epistemological.

There is one remaining fact about the mathematical structure of the
theory that anyone trying to come to an understanding about what it
says about the world has to grapple with. It is not a property of
Hilbert spaces, this time, but of the dynamics, the rules that
describe the trajectories that systems follow through the space. From
a physical point of view, it is far more worrisome than anything that
has preceded. For, it does much more than present difficulties to
someone trying to provide an *interpretation* of the theory, it
seems to point either to a logical inconsistency in the theory’s
foundations.

Suppose that we have a system S and a device S* which measures an
observable A on S with values {*a*_{1},
*a*_{2}, *a*_{3}...}. Then there is
some state of S* (the ‘ground state’), and some observable
B with values {*b*_{1}, *b*_{2},
*b*_{3}...} pertaining to S* (its ‘pointer
observable’, so called because it is whatever plays the role of
the pointer on a dial on the front of a schematic measuring
instrument in registering the result of the experiment), which are
such that, if S* is started in its ground state and interacts in an
appropriate way with S, and if the value of A immediately before the
interaction is *a*_{1}, then B’s value immediately
thereafter is *b*_{1}. If, however, A’s value
immediately before the interaction is *a*_{2}, then
B’s value afterwards is *b*_{2}; if the value of A
immediately before the interaction is *a*_{3}, then
B’s value immediately after is *b*_{3}, and so
on. That is just what it *means* to say that S* measures A. So,
if we represent the joint, partial state of S and S* (just the part
of it which specifies the value of [A on S & B on S*], the
observable whose values correspond to joint assignments of values to
the measured observable on S and the pointer observable on S*) by the
vector
|A=*a*_{i}>_{s}|B=*b*_{i}>_{s*},
and let "" stand in for the dynamical description
of the interaction between the two, to say that S* is a measuring
instrument for A is to say that the dynamical laws entail that,

|A=Intuitively, S* is a measuring instrument for an observable A just in case there is some observable feature of S* (it doesn’t matter what, just something whose values can be ascertained by looking at the device), which is correlated with the A-values of systems fed into it in such a way that we can read those values off of S*’s observable state after the interaction. In philosophical parlance, S* is a measuring instrument for A just in case there is some observable feature of S* whicha_{1}>_{s}|B=ground state>_{s*}|A=a_{1}>_{s}|B=b_{1}>_{s*}|A=

a_{2}>_{s}|B=ground state>_{s*}|A=a_{2}>_{s}|B=b_{2}>_{s*}|A=

a_{3}>_{s}|B=ground state>_{s*}|A=a_{3}>_{s}|B=b_{3}>_{s*}and so on.

^{[12]}

Now, it follows from (3.1), above, that there are states of S (too many to count) which are not eigenstates of A, and if we consider what Schrödinger’s equation tells us about the joint evolution of S and S* when S is started out in one of these, we find that the state of the pair after interaction is a superposition of eigenstates of [A on S & B on S*]. It doesn’t matter what observable on S is being measured, and it doesn’t matter what particular superposition S starts out in; when it is fed into a measuring instrument for that observable, if the interaction is correctly described by Schrödinger’s equation, it follows just from the linearity of the U in that equation, the operator that effects the transformation from the earlier to the later state of the pair, that the joint state of S and the apparatus after the interaction is a superposition of eigenstates of this observable on the joint system.

Suppose, for example, that we start S* in its ground state, and S in the state

1/2|A=It is a consequence of the rules for obtaining the state-space of the composite system that the combined state of the pair isa_{1}>_{s}| + 1/2|A=a_{2}>_{s}

1/2|A=and it follows from the fact that S* is a measuring instrument for A, and the linearity of U that their combined statea_{1}>_{s}|B=ground state>_{s*}+ 1/2|A=a_{2}>_{s}|B=ground state>_{s*}

1/2|A=This, however, is inconsistent with the dynamical rule for contexts of type 2, for the dynamical rule for contexts of type 2 (and if there are any such contexts,a_{1}>_{s}|B=b_{1}>_{s*}+ 1/2|A=a_{2}>_{s}|B=b_{2}>_{s*}

|A=Indeed, it entails that there is a precise probability of 1/2 that it will end up in the former, and a probability of 1/2 that it will end up in the latter.a_{1}>_{s}|B=b_{1}>_{s*}or

|A=

a_{2}>_{s}|B=b_{2}>_{s*}

We can try to restore logical consistency by giving up the dynamical
rule for contexts of type 2 (or, what amounts to the same thing, by
denying that there *are* any such contexts), but then we have
the problem of consistency with experience. For it was no mere
blunder that that rule was included in the theory; we *know*
what a system looks like when it is in an eigenstate of a given
observable, and we know *from looking* that the measuring
apparatus after measurement is in an eigenstate of the pointer
observable. And so we *know* from the outset that if a theory
tells us something else about the post-measurement states of
measuring apparatuses, whatever that something else is, it is
wrong.

That, in a nutshell, is the Measurement Problem in quantum mechanics; any interpretation of the theory, any detailed story about what the world is like according to quantum mechanics, and in particular those bits of the world in which measurements are going on, has to grapple with it.

If we don’t want to lose the distinction between pure and mixed
states, we need a way of representing the weighted sum of a set of
pure states (equivalently, of the probability functions associated
with them) that is different from adding the (suitably weighted)
vectors that represent them, and that means that we need either an
alternative way of representing mixed states, or a uniform way of
representing both pure and mixed states that preserves the
distinction between them. There is a kind of operator in Hilbert
spaces, called a **density operator**, that serves well in the
latter capacity, and it turns out not to be hard to restate
everything that has been said about state vectors in terms of density
operators. So, even though it is common to speak as though pure
states are represented by vectors, the official rule is that states
– pure and mixed, alike - are represented in quantum mechanics
by density operators.

Although mixed states *can*, as I said, be used to represent our
ignorance of the states of systems that are actually in one or
another pure state, and although this has seemed to many to be an
adequate way of interpreting mixtures in classical contexts, there
are serious obstacles to applying it generally to quantum mechanical
mixtures. These are left for detailed discussion in the other entries
on quantum mechanics in the Encyclopedia.

Everything that has been said about observables, strictly speaking,
applies only to the case in which the values of the observable form a
discrete set; the mathematical niceties that are needed to generalize
it to the case of **continuous observables** are complicated, and
raise problems of a more technical nature. These, too, are best left
for detailed discussion.

This should be all the initial preparation one needs to
*approach* the philosophical discussion of quantum mechanics,
but it is only a first step. The more one learns about the
relationships among and between vectors and operators in Hilbert
space, about how the spaces of simple systems relate to those of
complex ones, and about the equation which describes how
state-vectors move through the space, the better will be one’s
appreciation of both the nature and the difficulty of the problems
associated with the theory. The funny backwards thing about quantum
mechanics, the thing that makes it endlessly absorbing to a
philosopher, is that the more one learns, the harder the problems
get.

- Albert, D., 1992,
*Quantum Mechanics and Experience*, Cambridge, MA: Harvard University Press - Halmos, P., 1957,
*Introduction to Hilbert Space*, 2nd edition, Providence: AMS Chelsea Publishing

- Preskill, J., 1998, Quantum Computation (Lecture Notes for Physics 219, California Institute of Technology)

*First published: November 28, 2000*

*Content last modified: November 28, 2000*