Configuration space vs physical space

[Chrisc]

Demystifier, I think this points to a fundamental problem I've been struggling with
for a long time. What is measured "as" the properties of particles?
I'll paraphrase your OP for my purposes:
Entanglement is non-local in 3-space and local in configuration space.
Therefore non-locality is a problem in 3-space but not in configuration space.

This raises the question: is entanglement a state of particle properties endowed at creation or the state of configuration space on which we define their creation?

If entanglement can be understood NOT as a property of the particles in question but the geometry
of the configuration space on which they are created, non-locality is then NOT a condition of particle property
but a correlation between the configuration space on which the particles (with their corresponding "assumed" properties) are created and the configuration space on which the particles (and assumed properties) are measured.
Thus hidden variables do not exist as unknown qualities of particles nor can such metaphysical attributes account for the predictions of QM, but as correlations of configuration space between creation and detection entanglement is a local dynamic of the evolution of propagating fields.

I'm afraid my math skills are not up to the task, but as I see it, a particle cannot be defined as a finite point in 3-space except as the definition of detecting "in" 3-space the dynamics we set out to measure. The particle then is not defined by the dynamics measured any more than a baseball is defined by the wind blowing through the window it breaks.
In many ways this idea of measurement boils down to relativistic emergence - what you define as properties of a particle at creation evolves and emerges as (via relativistic field equations) what I define as the properties of detection.

 

[Hurkyl]

There's a glaring omission in the original question -- what are the observables?

 

[Chrisc]

Is this a question of distinguishing configuration space from physical space,
or is it a question of distinguishing degrees of freedom of action from the degrees of freedom a particle?
Two free particles in one dimension each have less degrees of freedom than one particle in two dimensions.
If the two particles express the greater probabilities corresponding to less degrees of freedom, the one particle in two
dimensions has less probabilities and greater degrees of freedom.
The former is the wave-function the latter the measurement as a measurement MUST define dimensionality.
By this analogy, uncertainty is simply a matter of the impossibility of a simultaneous confinement of degrees of freedom.

 

[Demystifier]

Maybe 3-space looks more physical to us than configuration space because physical space is 3-space. At least that's a possibility, isn't it?

It certainly is. In fact, this is the standard view.

If the real physical space is 3-space, then quantum nonlocality isn't a physical problem.

Quite the contrary, I think this is exactly why nonlocality is viewed as a problem.

 

[pellman]

Is this topic connected to the question:

How is the x in a quantum field $$\psi(x,t)$$ related to the N position operators $$\hat{x}_1,...\hat{x}_N$$ for a system of N particles?

The x in the quantum field refers to the coordinates on the manifold on which the field lives, while the position operators refer to configuration space.

 

[strangerep]

Is this topic connected to the question:

How is the x in a quantum field $$\psi(x,t)$$ related to the N position operators $$\hat{x}_1,...\hat{x}_N$$ for a system of N particles?

The x in the quantum field refers to the coordinates on the manifold on which the field lives, while the position operators refer to configuration space.

Is this topic connected to the question? I'd say yes. Taking it a bit further, there is a
"no-interaction" theorem of Currie, Jordan & Sudarshan. They draw a distinction between
"relativistic invariance" (meaning construction of a representation of the Poincare algebra),
and "manifest invariance" (meaning expressing all physical quantities in terms of 4D
spacetime and the particular ways in which things transform under changes of spacetime
reference frame). They show that these two approaches are not really compatible for
interacting multi-particle theories (hence the name "no-interaction theorem").

In orthodox QFT there's also something called the Reeh-Schlieder theorem which
exposes a related paradox.

IMHO, this "physical space" notion arises because an inertial observer's local
symmetry group is ISO(3,1). The set of measurements he/she can perform corresponds
(in the quantum sense) to the set of projection operators constructible in unirreps of
that group. I.e., the only measurements he/she can do are essentially equivalent to
"apply one of those projection operators". Hence the appearance of 3+1 physical space
for each observer - because he/she doesn't have the ability to wield a larger set of projection
operators. (This also leads to some of the puzzles that the physical spaces perceived
by different observers do not always coincide properly).

 

[ThomasT]

Originally Posted by ThomasT
If the real physical space is 3-space, then quantum nonlocality isn't a physical problem.

Quite the contrary, I think this is exactly why nonlocality is viewed as a problem.

If the real physical space is 3-space,
then if quantum nonlocality only 'occurs' in purely formal, nonphysical space,
then quantum nonlocality isn't a physical problem.

 

[Demystifier]

If the real physical space is 3-space,
then if quantum nonlocality only 'occurs' in purely formal, nonphysical space,
then quantum nonlocality isn't a physical problem.

The point is that quantum nonlocality occurs in a physical, not only purely formal, space. It is a measured effect.

 

[pellman]

there is a
"no-interaction" theorem of Currie, Jordan & Sudarshan. They draw a distinction between
"relativistic invariance" (meaning construction of a representation of the Poincare algebra),
and "manifest invariance" (meaning expressing all physical quantities in terms of 4D
spacetime and the particular ways in which things transform under changes of spacetime
reference frame). They show that these two approaches are not really compatible for
interacting multi-particle theories (hence the name "no-interaction theorem").

In orthodox QFT there's also something called the Reeh-Schlieder theorem which
exposes a related paradox.

This sounds very interesting. Thanks for sharing.

 

[ThomasT]

The point is that quantum nonlocality occurs in a physical, not only purely formal, space. It is a measured effect.

You might say, depending on experimental design and observed instrumental behavior, that IF something is propagating FTL in physical 3-space then there are some limits on that. But there's no reason that I know of to assume that something IS propagating FTL in physical 3-space. All that's known is that quantum entanglement, quantum wavefunction collapse, and quantum nonlocality are creatures of the qm formalism, and that the formalism is a probability calculus which ultimately produces probability densities arising from functions which describe 'propagations' in a nonphysical space.

 

[debra]

The relevant property in configuration space is that of state correlation (and not so much else it seems) which when applied to physical space requires an explanation in terms of retained state relationships applicable from creation time onwards (avoiding local variables etc), or is somehow maintained in some synchronous way. e.g. by a shared timing function underlying space time (physical space) but applies in the configuration space.

 

[pellman]

Is this topic connected to the question:

How is the x in a quantum field $$\psi(x,t)$$ related to the N position operators $$\hat{x}_1,...\hat{x}_N$$ for a system of N particles?

The x in the quantum field refers to the coordinates on the manifold on which the field lives, while the position operators refer to configuration space.

Just to elaborate a bit further: In regular quantum mechanics, the dynamical equations for a system of N particles can be expressed in terms of any number of different sets of 3N generalized coordinates $$\hat{q}_1,...\hat{q}_{3N}$$. Each set, if we write down the correct Hamiltonian, is just as valid as the others and gives the right answers.

This freedom has nothing to do with different representations, e.g. the momentum representation. Each set of 3N generalized coordinates is an equally valid "position" representation.

Yet there is an especially unique set of generalized coordinates (up to global transformations) $$\{\hat{q}_1,...\hat{q}_{3N}\}=\{\hat{x}_1,...\hat{x}_N\}$$ which we say refer to the "positions of the particles". What is it that is special about this set of coordinates? I think we can consider this to be a first stab at a mathematical formulation of the OP question.

On the other hand, there is the x which appears in a quantum field $$\psi(x)$$. I know enough QFT to know that $$\psi(x)$$ is associated with the position representation and that we can transform to, say, the momentum representation and find the associated field $$\phi(p)$$, but what is the analogy in QFT to a set of generalized coordinates as described above for QM?

 

[strangerep]

Yet there is an especially unique set of generalized coordinates (up to global transformations) $$\{\hat{q}_1,...\hat{q}_{3N}\}=\{\hat{x}_1,...\hat{x}_N\}$$ which we say refer to the "positions of the particles". What is it that is special about this set of coordinates?

I don't think it's "special" in itself. Only the full specification of the dynamics with its
degrees of freedom is physically relevant.

On the other hand, there is the x which appears in a quantum field $$\psi(x)$$. I know enough QFT to know that $$\psi(x)$$ is associated with the position representation and that we can transform to, say, the momentum representation and find the associated field $$\phi(p)$$, but what is the analogy in QFT to a set of generalized coordinates as described above for QM?

In QFT (Fock space), there are operators to create/annihilate particles with given
momenta, spin, etc. One can inverse-Fourier transform to a position-like basis, or indeed one
can start (in axiomatic QFT) from an irreducible set of field operators defined on Minkowski
space. The generalized set of coordinates you mentioned is loosely analogous to the tensor
products of 1-particle Hilbert spaces used in the construction of Fock space. In both cases,
we build up larger and larger dynamical systems via tensor products.

But either way, you run into the embarrassing Reeh-Schlieder paradox.

 

[Demystifier]

But either way, you run into the embarrassing Reeh-Schlieder paradox.

What is Reeh-Schlieder paradox?

 

[atyy]

So, are you saying that the known physical laws would allow the existence of living beings that would think that they live in, e.g., 5 "physical" dimensions? I don't think so.

Not sure what nughret was thinking, I was thinking along the lines of a soundwave hitting the ear - it's 1D as a function of time. But what we hear is eg. 3D in a keyboard fugue, or 25D in a Strauss tone poem. Do people really look out and see 3D physical space? Or point particles for that matter?

 

[QuantumBend]

What is Reeh-Schlieder paradox?

Reeh Scleider paradox for Quantum Feild Theory: not operator IN particle but field observe operator ONLY.
- no good.

 

[Demystifier]

not operator IN particle but field observe operator ONLY.

I do not understand this sentence.

 

[ThomasT]

Demystifier, I looked at several papers dealing with Reeh-Schlieder theorem (or paradox or property), and can confidently say that I don't fully understand its significance ... yet.

Now I'm thinking that maybe the consideration of your thread is a bit over my head for the foreseeable future. But thanks for tolerating my comments.

One parting comment, before retiring to the peanut gallery (where, of course, I'll continue to follow others comments, and look stuff up).

Do people really look out and see 3D physical space? Or point particles for that matter?

Not point particles. But events in the 'space' of our sensory perception are communicable using 3 spatial and 1 time dimension. It remains to be seen, literally, if there's any physical space other than this.

 

[strangerep]

[...] But either way [in orthodox QFT] you run into the
embarrassing Reeh-Schlieder paradox.

What is Reeh-Schlieder paradox?

It's a theorem applicable to axiomatic QFT. Having started with an
irreducible set of causal fields over Minkowski space carrying a +ve energy
unirrep of the Poincare algebra, the R-S thm then essentially is this:

Let A,B be two disjoint spacelike-separated regions in Minkowski
spacetime. Given only knowledge of the field configuration on region A
it is possible to reconstruct the field on region B. This is embarrassing
because stuff happening in region A should physically have nothing to
do with region B. (I used the word "paradox" because we started with
a supposedly causal theory, yet we derived this theorem, but the
phrase "physical contradiction or puzzle" might be more appropriate.)

This can be restated in various ways. E.g., "local operations applied to
the vacuum state can produce any state of the entire field" [1].
Or "The R-S thm asserts the vacuum and certain other states to be
spacelike superentangled relative to local fields". [2]

It's a rather controversial subject. (E.g., see Wiki's entry.)

[...] not operator IN particle but field observe
operator ONLY

I do not understand this sentence.

I'm not sure I do either. Possibly QuantumBend was pointing out
that R-S is applicable to QFT, not relativistic particle theory.

---------------------------------
Refs:

[1] Halvorson, "Reeh-Schlieder defeats Newton-Wigner ..."
available as quant-ph/0007060
(see also refs therein)

[2] Fleming, "Reeh-Schlieder meets Newton-Wigner"
Phil Sci 67 (proceedings), S495-515
http://philsci-archive.pitt.edu/archive/00000649/00/RS_meets_NW,_PDF.pdf

 

[Demystifier]

Thanks, strangerep.
But I still cannot say that I understand it.
For example, what if we replace a continuous space by a lattice, is the RH theorem still valid then?

 

[jambaugh]

I have contemplated this issue for many years myself. Here are some of my thoughts:

Recall that in non-relativistic theory time is not an observable but rather a parameter. In the unification of space-time we can choose to re-interpret time as an observable or lose the "observable" status of spatial coordinates and treat them too as parameter, i.e. no different than abstract configuration coordinates in phase space.

I think this second is the correct view and think breaking Born reciprocity is a good thing. Once we move to field theory this transition is complete. The observables are field values (particle type and number charge etc) at a given coordinate position. Thus coordinates are numbers we put on our measuring devices.

This to some is a problem given GR which in the current geometric formulation treats gravitation as curved space-time geometry. However if we read the Equivalence Principle correctly (as I see it) then it is not that "gravity is just geometry" but rather that we only see dynamic evolution of test particles and so the boundary between gravity and geometry is indistinguishable. We can vary our choice of space-time geometry and introduce a "physical" force of gravity and not see any difference in predictions. I think this means rather that it is the geometry which is "not physical" rather than the gravitational force.

I also think failure to see this view has hampered quantum grav. research.

 

[QuantumBend]

I'm not sure I do either. Possibly QuantumBend was pointing out that R-S is applicable to QFT, not relativistic particle theory.

I saying: In QFT we say NOT 'particle here', we saying 'OPERATOR who observe particle here'. You know this. When no observe - no particle, no history, big one area. 3 spaces no good.

This saying, I know in complex mathematic. Why?
https://www.physicsforums.com/showthread.php?t=270084&highlight=photon+exist
Big one, no good - Reeh-Schlieder do this.
https://www.physicsforums.com/showthread.php?t=282289&highlight=single+photon

 

[strangerep]

what if we replace a continuous space by a lattice, is the RH theorem still valid then?

Hmmm, I'm not sure.

I've just had a look at the proof of the RS theorem given in Appendix 4
of Lopuszanski [1]. It relies on some functional-analytic results, together
with an extension to complex spacetime variables to ensure a certain
integral is meaningful.

A strictly discrete spacetime is very different from the foundations
used in axiomatic QFT.

---------------
[1]: Lopuszanski, "An Introduction to Symmetry [...] in QFT",
World Scientific, ISBN 9971-50-161-9.

 

[turin]

I got tired of reading halfway through the thread, so I appologize if I am repeating anyone.

I am considering the very original post, with the Hamiltonian H=p1^2+p2^2. I see two possibile distinguishing features between particle degrees of freedom vs. spacel degrees of freedom, but both of them require extending to other considerations besides the Hamiltonian.

1) The statistics of combining two space degrees of freedom for a single particle is trivial. The statistics of combining two particle degrees of freedom in a 1-D space is nontrivial. E.g., a single fermion in 2-D space doesn't care that it is a fermion, and, in particular, p1=p2 is allowed. Two fermions in a 1-D space care that they are fermions, and, in particular, p1=p2 is not allowed.

2) The topology of 2 space degrees of freedom for a single particle is different than the topology of 2 particle degrees of freedom in a 1-D space. E.g., for a single particle in 2-D space, I could say that one unit of momentum in the p1 direction is \sqrt{2} units of momentum away from one unit of momentum in the p2 direction in momentum space. However, I think it might be less meaningful/physical to talk about the amount of separation between one unit of momentum for particle 1 and one unit of momentum for particle 2.

 

[RedX]

I thought in classical mechanics all phase spaces with the same # of dimensions are the same. Nothing weird happens: it's just a flat space. Specifying a Hamiltonian just specifies one of many different types of canonical transformations you can perform among the coordinates of the space.

If the phase space is all the same I don't see how you can ask if the configuration space is different because a single rule for converting a phase space into a configuration space should convert the same phase space into the same configuration space?

Anyways, this thread reminds me of a question I have about classical mechanics. In the Hamiltonian formulation, areas in phase space are conserved, i.e., the divergence of the phase space velocity is zero. In the Lagrangian formulation, the phase space is coordinates and their velocity, but the divergence of the phase space velocity is not in general zero. Does this mean that determinism is violated in the Lagrangian formulation, since 10 initial phase states will not go into 10 final phase states in general?

 

[Demystifier]

Anyways, this thread reminds me of a question I have about classical mechanics. In the Hamiltonian formulation, areas in phase space are conserved, i.e., the divergence of the phase space velocity is zero. In the Lagrangian formulation, the phase space is coordinates and their velocity, but the divergence of the phase space velocity is not in general zero. Does this mean that determinism is violated in the Lagrangian formulation, since 10 initial phase states will not go into 10 final phase states in general?

If something is not conserved, it does not mean that it does not behave deterministically. So it certainly does not mean that determinism is violated in the Lagrangian formulation.

 

[RedX]

If something is not conserved, it does not mean that it does not behave deterministically. So it certainly does not mean that determinism is violated in the Lagrangian formulation.

Well, if you have a collection of 5 initial states, then after some time, there should only be at most a collection of 5 final states. If you have a collection of 6 final states, that means the Hamiltonian took one state, and outputed two states, violating determinism. So areas in phase space should never expand, so that if your area is 5 states, then it ought to be no more than 5 states after time translation. Hamilton's q-p space is conserved after evolution in time, but Lagrange's q-dq/dt space is not conserved.

 

[jambaugh]

I thought in classical mechanics all phase spaces with the same # of dimensions are the same. Nothing weird happens: it's just a flat space. Specifying a Hamiltonian just specifies one of many different types of canonical transformations you can perform among the coordinates of the space.

Just a comment...

In the more advanced formulation, one works in an extended phase-space which is flat as you say but not all functions of the canonical coordinates and momenta are observables. Some of the dimensions are gauge degrees of freedom. By imposing gauge constraints you pick out a curved sub-manifold of the extended phase-space, define a Dirac bracket instead of the Poisson bracket with which canonical transformations map this sub-manifold onto itself, and thereby work in a curved physical state manifold. What's more the dimension need not be even.

How this relates to the OP is the observation that configuration space is in this general case may be a mixture of physical and gauge degrees of freedom (as is the canonical momentum space). It is not until the gauge constraints are chosen that physical degrees of freedom are well defined.

Note that (e.g. in electrodynamics) it is the canonical momenta [itex] P = p + ieA(x)[/tex] which are the "flat" coordinates and then the physical momenta [itex]p = P - ieA(x)=mv[/tex] which one defines relative to P when one fixes the gauge. (I may have the +/- signs mixed up but that's a matter of convention.) The p's live on a curved sub-manifold of phase space.

In a more general case one may introduce a U(1) gauge phase as another canonical coordinate in configuration space with dual momentum corresponding to the particle's charge. The phase connection A then defines what mixtures of canonical coordinates correspond to physical coordinates. The hamiltonian is greatly simplified but the physical forces are due to gauge constraints. It is analogous to gravitation via geometry in GR.

[Edit: A good reference is "Quantization of Gauge Systems" by Henneaux and Teitelbaum ]

 

[Fra]

Reflection on the puzzle

I just bumped into this thread and didn't follow it from start, but relating to Demystifiers original question on the differentiation between what "space" is "more physical". I thikn it's an interesting question. This is related to something I'm also pondering. It also connects to Hurky's remark, about what are the observables? Ie. what questions are more "physical"? I think the mathematical view here is not helping.

I am still working on this in a larger context but to me, this question is really deeply entangled with the microstructure of the observer, as well as the concept of inertia of the structure, and I like to think that the answer to why certain microstructures are favoured (say a 3D space) lies at the level of evolving relations.

Someone pondered the idea that, would the laws of physics "allow" an observer who thinks he lives in a 5D space? I think it does. But the question is, what would happen to such an observer, would it be fit and stable? :) Probably not, to me I think an analogous question is what is the evolutionary mechanism that can explain the plausability of the emergent common structures we see.

I am attacking this from the point of view of picturing interacting microstructure-systems (which to me is the abstraction of "an observer", but which well may represent a physical system, say a particle with given properties, properties that are implicit in the makeup of the microstructure), and the trick would be that the interaction between the systems implies a selective pressure that causes evolution of the structures. Thus each propertiy such as dimensionality is seen as a relation between the systems environment, and has no universal sense beyond it's current evolutionary status.

I'm trying to get my head around exactly how this interacting driven evolution produces the basic structures we konw which, would be spacetime and basic properties of the simplest possible observers (elementary particles) that contains the four forces. So I think the propertis of the simplest structurs around, and the emergence of space go hand in hand.

Any attempt to study one, idealised and disconnected from the other doesn't make sense to me.

So I agree it's a puzzle, and I have no answer either, I only have at least for myself a strategy and plan I'm working along the spirit explained above.

/Fredrik

 

[Dmitry67]

Interesting thread.
Assuming Max Tegmarks MUH, if 2 systems are isomorphic then they are the same. So, if we can map our physical space to configurational space then both are physical... or configurational...

 

[Ken G]

I actually take a perspective which is more or less opposite to that of Tegmark's. He seems to feel that the purpose of mathematics is to establish the true ontologies of reality, so if two mathematical descriptions are equivalent, then the true ontologies are the same-- even if they sound different before we understand the mathematical equivalence. My approach is that the goal of physics never was to find true ontologies, so the fact that mathematical ontologies are "always true" (within the mathematical theory involved) demonstrates the different goals of math and physics. Physics borrows mathematical ontologies for specific, contextually dependent purposes, and these ontologies are not unique and are not meant to be unique. There is just no such thing as a "true ontology" in physics, and there is no need for one. We don't use true ontologies, we use effective or useful ontologies, and this is quite demonstrably true about physics. So I don't think we should lose any sleep over what is the "true ontology" that should be associated with a particular Hamiltonian, or model of any kind. A model is a model, not a true ontology, and works for whatever it works for.

That doesn't mean I don't think the OP question is interesting-- indeed, one of the most interesting things about it, in my view, is how it can be used as a device to establish this point.

 

[bohm2]

So, if we can map our physical space to configurational space then both are physical... or configurational...

I thought that can't be done because there are so many different ways of doing it and the choice seems arbitrary. As I understand it, Lewis does attempt to do that here:

But suppose instead that we take seriously the idea of a configuration space as a space of configurations-that is, a space which is intrinsically structured as N sets of three-dimensional coordinates. Mathematically, this is not hard to do. Instead of modeling the space as an ordered 3N-tuple of parameters, <x₁, x₂, x_3N> we model it as an ordered N-tuple of ordered triples:

<<x₁,y₁,z₁>, <x₂y₂,z₂>,...<x_N,y_N.z_N>>

And rather than specifying the coordinates by choosing 3N axes, we choose 3-the x, y and z axis, which are the same for each triple. That is, x₁ through x_N pick out points on the same axis, and similarly for y and z. Then the wavefunction can be regarded as a function of these parameters-as a mathematical entity inhabiting a (3 x N)-dimensional configuration space, rather than a 3N-dimensional plain space. And the basic thesis of wavefunction realism is that the world has this structure-the structure of a function on (3 x N)-dimensional configuration space. Given that configuration space has this structure, then an Albert-style appeal to dynamical laws to generate three-dimensional appearance is impossible, but it is also unnecessary. It is impossible because the dynamical laws take exactly the same form under every choice of coordinates (as they should), so no choice makes the dynamical laws simpler than any other. But it is unnecessary because the outcome of that argument-that the coordinates are naturally grouped into threes is built into the structure of reality, and hence doesn’t need to be generated as a mere appearance based on the simplicity of the dynamics.

http://philsci-archive.pitt.edu/8345/1/dimensions.pdf

But others like Monton question this:

http://spot.colorado.edu/~monton/BradleyMonton/Articles_files/qm%203n%20d%20space%20final.pdf
http://spot.colorado.edu/~monton/BradleyMonton/Articles.html (see "Against-3N -dimensional space")

Having said that I'm really confused about this whole topic because is anyone clear on what "dimensionality" means in configuration space?

 

[qsa]

The question (or puzzle) that I want to pose ...

Maybe you can answer you own question. What was the original idea and intend for introducing the configuration space in classical physics ? Was there a price to pay?

 

[Ken G]

And why are we limiting the discussion to configuration vs. Euclidean space? Why not phase space? If I specify the configuration of N particles, and their Hamiltonian, I still don't know enough about them to predict what happens, so that "must not be the reality" either. Specifying a "rate of change of configuration" for each particle is a bizarre way to provide realism to the picture, it would be much more natural to use a 6N dimensional phase space, and call that the reality.

Of course, as I said above, I think calling any of these things reality is a kind of breakdown of sound scientific thinking. The purpose of science is to replace reality with models of reality that achieve various purposes, and it is both fruitless and unnecessary to ask, which one is the "real reality." For example, consider this from the Lewis quote just above:" And the basic thesis of wavefunction realism is that the world has this structure-the structure of a function on (3 x N)-dimensional configuration space." I would have to say that if that is realism, then he can keep it-- it sure doesn't sound like science to assert that the world has a certain structure. What sounds like science is saying "let us provisionally enter into a state of imagination that the world has this structure, because it serves us in the following ways." Stated like that, doesn't the whole issue just dissipate in the way it should?

 

[Demystifier]

What was the original idea and intend for introducing the configuration space in classical physics ?

Mathematical elegance.

Was there a price to pay?

Yes, obscure visualization in the 3-space.