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by unknown author.

It is commonly asserted that in general relativity there is no absolute

simultaneity. On the other hand, it is asserted that, based on the traveling

time of light, we see the Sun as it was 8 minutes ago and the Andromeda nebula

as it was 2.5 million years ago. This seems to conflict with each other -

apparently we have no diffeomorphism invariant way of assigning a relative time

to a distant object. Let us take a closer look at the issues involved.

The invariant way of defining present is to say that x and y are present if the

two points are in a spacelike relation, and to say y was earlier (or later) than

x if y lies in or on the past (or future) light cone. Thus the present is

well-defined as the complement of the closed light cone.

Now suppose that you look at the sun. If one is really pedantic, one would have

to say that you see the sun in your eye, as a 2D object, and not out there in

3D. But we are accustomed to interpret our sensations in 3D and hence put the

sun far away but into the here.

In general relativity, one goes a step further. One thinks in terms of the 4D

spacetime manifold and places the sun there. Calculating the length of the

geodesic gives a value of 0, so the sun is not in your present. Consideration of

the sign of the time component in an arbitrary proper Lorentz frame, one finds

that the sun is in your past, as everything you observe.

But the amount of invariant time passed, as measured by the metric, is zero.

This looks like a paradox. What happened with the claimed 8 minutes?

The answer is that the metric time is not the right way to measure time. It is

the only time available in a Poincare-invariant flat universe, or in a

diffeomorphism invariant curved universe. An empty universe where only

noninteracting observers move has no notion of simultaneity.

But a matter-filled, homogeneous and isotropic universe generally has one,

defined by the rest frame of the galactic fluid with which general relativity

models cosmology. Since the fluid breaks Lorentz symmetry (except in very

special cases, which are ruled out by experiment) it creates a preferred

foliation of spacetime. This foliation gives a well-defined cosmic time, when

scaled to make the expansion of the universe uniform. (Actually there are

several natural scalings = monotone transformations of the time parameter; see

Section 27.9 in Misner/Thorne/Wheeler, so cosmic time without a reference to the

scale used is ambiguous.)

This cosmic time figures in all models of cosmology. The values commonly talked

about when quoting times for cosmological events, such as the date of the big

bang or the time a photon seen now left the Andromeda nebula, refer to this

cosmological time. Concept of Now : Time is passing. This means that what is

''now'' in our subjective experience changes. But there is no concept of ''now''

in physics.

Classical nonrelativistic mechanics does not know the concept of now. One

declares some time to be ''now'' - but which time one declares to be ''now'' is

completely subjective (i.e., in different situations it will be declared

differently). Similarly, one declares some position to be ''here'', but which

position you declare to be ''here'' is completely subjective, in the same sense.

Since different people assign at different times a different meaning to the time

called ''now'' and the position called ''here'', these words have - from an

objective point of view - simply the meaning of a variable denoting time and

position, respectively. Classical relativistic mechanics does not know the

concept of now, either, but things change a little: Here one declares some event

(= spacetime point) to be ''here and now'' - but which event one declares to be

''here and now'' is completely subjective. Nonrelativistic quantum mechanics

treats time completely differently from space (time is a parameter, space

coordinates are operators), and introduces stochastic elements into the

dynamics. but with respect to ''here'' and ''now'', the situation is identical

with that in the classical nonrelativistic case.

Relativistic quantum mechanics restores the treatment of space and time on equal

footing (space annd time coordinates are parameters), and introduces stochastic

elements into the dynamics. But with respect to ''here and now'', the situation

is identical with that in the classical relativistic case.

Once one has chosen ''here'' and ''now'', respectively ''here and now'', it

serves as origin of the tangent hyperplane, in which localized, flat physics can

be done, reflecting faithfully what happens in a neigborhood of the spacetime

point. This is the domain of relativistic quantum field theory. Time in Q.M Time

in Quantum Mechanics : In the traditional formulation of quantum mechanics, time

is not an observable. Nevertheless it can be observed...

In the Schroedinger picture, the state is defined at fixed times, which

distinguishes the time. In this picture, time measurement is difficult to

discuss since the time at which a state is considered is always sharp.

In the Heisenberg picture, time is simply a parameter in the observables, and

therefore also distinguished, but in a different way. Parameters are in fact

just continuous indices and not observables. As 3 is not an observable while p_3

is one, so t is not an observable but H(t) is one. Observables have at _each_

time an expected value; the moment of time (''now'') is not modelled as

observable.

But what can be modelled is a clock, i.e., a system with an observable which

changes with time in a predictable way. If the observable u(t) of a system

satisfies ubar(t) := <u(t)> = u_0 + v (t - t_0) (v nonzero) (*) with

sufficient accuracy, one has a clock and can find out by means of <u(t)> how

much time T = Delta t passed between two observed data sets. This is also the

usual way we measure time in classical physics.

Of course, to be a meaningful time measurement, T must be large enough compared

with the intrinsic uncertainty Sigma_T := |v^{-1}| sigma(u(t)). Here sigma(u(t))

= sqrt(<(u(t)-ubar(t))^2>) is the standard deviation in the properly calibrated

(quantum mechanical) state <.>. If (*) has significant errors then Sigma_T is of

course correspondingly larger.

In relativistic quantum field theory (which in its covariant Version can only be

formulated in the Heisenberg picture), the 1-dimensional time t turns into the

4-dimensional space-time position x. Now x is a vector parameter in the

observables (fields), and hence is not an observable. Space and time are now on

the same level (allowing a covariant point of view), but both as

non-observables. The observables are fields; positions and times of particles

are modelled by unsharp 1-dimensional world lines characterized by a high

density of the expectations of the corresponding fields. (Think of the trace of

a particle in a bubble chamber.)

For position and time measurement, one now needs a 4-vector field u(x) with

<u(x)> = u_0 + V (x - x_0) and a nonsingular 4x4 matrix V, and the intrinsic

uncertainty takes the form Sigma_T := sigma(V^{-1}u(x)) with sigma(a(x)) =

sqrt(<(a(x)-abar(x))^*(a(x)-abar(x))>), abar(x)=<a(x)>.

Conclusion: In nonrelativistic quantum mechanics, time is always measured

indirectly via the expectations of distinguished observables of clocks in

calibrated quantum mechanical states. In relativistic quantum field theory, the

same holds for both position and time.

However, this analysis works only when one assigns to single clocks a

well-defined state, hence assumes a version of the Copenhagen interpretation.

From the point of view of the minimal statistical interpretation, one needs in

contrast a whole ensemble of identically prepared clocks to measure time...

Note that in relativistic quantum mechanics, a single particle is described (in

the absence of an external field) by an irreducible representation of the

Poincare group. Here only the components of 4-momentum and the 4-angular

momentum are observables. From these, one can reconstruct observer-dependent

3-dimensional (Newton-Wigner) position operators satisfying canonical

commutation rules, but not a time operator.

Time and Space: In QED (the most accurate theory we have), space and time are

parameters ranging continuously in R^4, coordinatizing the fields that contain

the physical information. These coordinates have no absolute meaning since

changing them by means of a Poincare transformation (a combination of

translation + rotation + Lorentz boost) does not alter the physics.

The resulting affine pseudo-metric space, called Minkowski space, is (in QED)

absolute and physically meaningful: All Poincare invariants expressible in terms

of the fields can (in principle) be determined objectively. In particular, this

holds for the Minkowski distance between space-time points that can be defined

in terms of the fields. Such space-time points include for example all positions

of stars, which are local maxima of field intensities in the backward light cone

of an observer at a particular time, singled out objectively by appropriate

observables.

According to established physics, a real observer is a macroscopic object with

the capacity to record information. The recording process is described by means

of irreversible thermodynamics. In particular, observers can be described to

good accuracy classically, in terms of their associated macroscopic observables.

These are expectation values of corresponding aggregated microscopic variables,

behaving essentially classically according to Ehrenfest's theorem. Large objects

such as stars can similarly be described by their associated macroscopic

observables. The position of an observer and the objects it observes changes in

time, defining their trajectories = world lines (apart from a global Poincare

transformation). This change is (on the macroscopic description level

appropriate for observers) continuous. (The world lines get fuzzy as one

focusses on smaller and smaller details, and become undetermined in principle

when the scale is reached where quantum effects dominate. Indeed, the Heisenberg

uncertainty principle forbids well-defined trajectories of arbitrary accuracy.)

For example: The observer might be the Mount Palomar observatory, at a fixed

time t. (This may be defined locally, say, ''one year after it was built'', a

property that may be encoded in terms of QED using known physics.) The

observer's past light cone cuts out from 4-space a 3-dimensional manifold, which

intersects the world lines of the objects observed at definite points (within

the accuracy of the whole construction) - the positions x(t) of the visible

stars at time t. This is consistent with how astronomical positions are

determined.

On the level of QED - and even one level below, in the standard model - space

and time are not quantized in any sense. What is quantized are the observable

fields, in dependence on time and position.

The situation changes slightly if we consider quantum gravity, thought to be

relevant on the smallest significant scales of our universe. It is tentatively

explored by current physicists, but without any definitive results so far.

Judging from general relativity (which must be the classical limit of any

meaningful quantum gravity), Minkowski space is now replaced by a

pseudo-Riemannian manifold, the translation subgroup of the Poincare group is

extended to the diffeomorphism group (or, because of anomalies, perhaps only to

the volume-preserving subgroup). Therefore, only the invariants under this

bigger group are physical. This still includes geodesic distance, so that the

picture painted above remains valid in Riemann normal coordinates for any

particular observer.

There is an intrinsic asymmetry between observed position and time - even in the

classical relativistic case!

Whether measured or not, a state is _always_ a state at a particular time t.

Thats why we write state densities as rho(t) and wave functions as psi(t), and

have a dynamical equation to tell how the state changes with time.

Note that in a relativistic theory, position becomes like time rather than time

like position: Instead of trajectories depending on time we have fields

depending on space and time. Note that absolutely precise particle positions for

multiparticle systems don't make relativistic sense - position becomes an

intrinsically smeared concept, even classically!

What one can have consistently in relativity is only relative positions of one

particle with respect to a frame attached to a particular particle and its world

line - in the case of the GPS this ''particle'' is the earth. These positions

are again time-dependent. In nonrelativistic quantum mechanics, a unique

position operator is defined only for a system consisting of a single particle

alone in the universe, in an observer-dependent coordinate system. (The observer

must be outside this mini universe.)

For an N-particle system, one has N position operators. If time were like

position, each particle would have its own time, which would make the concept of

time meaningless. But time has meaning; so this is not an option. operators. If

time were like position, each particle would have its own time, which would make

the concept of time meaningless. But time has meaning; so this is not an option.

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Oliver Thewaltthanks to Moinak Banerjee and fb.com/PHMATH !