I think your question is perhaps not very clear but I think I know what you might be asking.

As Stam Nicolis hinted at, configuration space is not the same as physical space. Physical space is concerned with spatial and temporal dimensions. Configuration space is defined by the parameters of a system.

Quantum mechanics is often formulated in terms of complex number and the wavefunction is assumed to be complex. Sakurai for example looks at the Stern-Gerlach experiment and concludes we need complex numbers to describe the experiment. I disagree. It is true that the three vectors in the three spatial dimensions are not sufficient to explain the experiment and so a larger vector space is required. However introducing i the unit imaginary is not the only way to do this.

Geometric algebra provides a way of multiplying vectors to to obtain higher grade objects called bivectors and trivectors. There represent oriented planes and oriented volumes. In 3D, geometric algebra gives us an 8-dimensional algebra consisting of a scalar, 3 vectors, 3 bivectors, and a trivector. all of this exists within 3 dimensional space. To see this you can consider the scalar to be a point, the 3 vectors to be the three axes, the three bivectors to be three perpendicular planes, and the trivector to be a volume element. This view is nice and geometric.

It is possible to describe the Stern-Gerlach experiment in a complete real space without the need to introduce complex numbers by making use of the three basis vectors and the three bivectors. this gives a real and geometric description of the experiment. Other (such as Doran and Lasenby) have also looked at quantum mechanics from a geometric algebra point of view and have also found that the i's in quantum theory can easily be removed.

So to answer the title of your question: yes, I believe it is possible to have a description of quantum mechanics in real space.

Regards

Dear Faramarz,

I'm not sure that I understand your question. Would you be equally concerned about a probability distribution on a 3N-dimensional configuration space? Or are you puzzled about the physical interpretation of a wavefunction?

Best regards,

David

The wavefunction is defined in phase space, not spacetime and the square of its modulus defines a probability density in phase space, the space of states of the system.  The spacetime description of quantum phenomena was defined by Dirac in 1933 and reformulated and worked out by Feynman in the 1940s (and is called now the path integral formalism). In this context quantum mechanics can be consistently defined as a quantum field theory in 0+1 dimensions (or one Euclidian dimension), where the fields depend on time only and the spacetime symmetry is time translations, for one field and the product of time translations and spatial rotations for more fields. This approach can, also, be generalized for the relativistic particle and was studied by many people in the 1970s and 1980s.

Hello Faramarz Rahmani,

I got your question and its a valid question.

I was able to answer your question using 3 D complex plane in paper attached with this post.

I explained why more than 3 dimensions come into play.

Regards,

Bhushan Poojary

Thesis Certainty Principle Using Complex Plane

I think your question is perhaps not very clear but I think I know what you might be asking.

As Stam Nicolis hinted at, configuration space is not the same as physical space. Physical space is concerned with spatial and temporal dimensions. Configuration space is defined by the parameters of a system.

Quantum mechanics is often formulated in terms of complex number and the wavefunction is assumed to be complex. Sakurai for example looks at the Stern-Gerlach experiment and concludes we need complex numbers to describe the experiment. I disagree. It is true that the three vectors in the three spatial dimensions are not sufficient to explain the experiment and so a larger vector space is required. However introducing i the unit imaginary is not the only way to do this.

Geometric algebra provides a way of multiplying vectors to to obtain higher grade objects called bivectors and trivectors. There represent oriented planes and oriented volumes. In 3D, geometric algebra gives us an 8-dimensional algebra consisting of a scalar, 3 vectors, 3 bivectors, and a trivector. all of this exists within 3 dimensional space. To see this you can consider the scalar to be a point, the 3 vectors to be the three axes, the three bivectors to be three perpendicular planes, and the trivector to be a volume element. This view is nice and geometric.

It is possible to describe the Stern-Gerlach experiment in a complete real space without the need to introduce complex numbers by making use of the three basis vectors and the three bivectors. this gives a real and geometric description of the experiment. Other (such as Doran and Lasenby) have also looked at quantum mechanics from a geometric algebra point of view and have also found that the i's in quantum theory can easily be removed.

So to answer the title of your question: yes, I believe it is possible to have a description of quantum mechanics in real space.

Regards

I'm somewhat familiar with  geometric algebra. I believe that configuration space is only a mathematical tool. We do not have a clear idea of the quantum world. We focus only on the results and mathematical models. ...

Yes, you can constuct a new type of QM using real-valued function, but this theory will contain very little results.

As per my current findings it seems early scientist used imaginary number to get there equations right.

But it seems now to me that position,momentum,energy are in complex form , hence imaginary number came in QM.

Thesis Certainty Principle Using Complex Plane

Thesis Energy equation in complex plane

Yes.  Quantum mechanics can indeed be reformulated entirely in terms of real-valued probabilities of measurement outcomes. For example, see:

W. Wootters, "Quantum Mechanics without Probability Amplitudes", FOUNDATIONS OF PHYSICS 16, 391 (1986).

L. Hardy, "Quantum Mechanics from 5 Reasonable Postulates", http://arxiv.org/abs/quant-ph/0101012v4

J. Barrett, "Information processing in generalized probabilistic theories", PHYSICAL REVIEW A 75, 032304 2007

This should not be surprising since QM, like any physical theory, is ultimately a theory of (real) observables. What is notable is that elements of the real-valued formulation are related to the elements of the traditional formulation simply by a linear map, and that most of the features of the traditional theory are retained in the real-valued theory.

Answer is Yes. Quantum mechanics is nice and with new insights formulated in the Pauli algebra (3D euclidean Clifford algebra over reals, Cl3). The imaginary unit is replaced by pseudoscalar (element of the center of algebra). Geometric interpretations are astonishing.  

This is not strange, Pauli matrices are just the 2D matrix representation of Cl3, they represent a unit vectors. Physicists know that one can multiply Pauli matrix by imaginary unit, but that means to multiply vector by pseudoscalar, resulting generally in an even element of algebra (spinor!). Such an object rotates and dilates objects, and that activity is geometrically completely clear, You can easy to draw it! Multiplying matrices one can see nothing, but if You are working with vectors, that is geometrically clear.

So, there is no need for the old ordinary imaginary unit, it is too poor in capabilities to live in 3D. But pseudoscalar, that beast is from 3D and it is very, very powerful!     

It is really unbelievable to me how many people like to stick with old, well used to it, concepts.

Geometric algebra is not just an another "interesting peace of mathematics" . It is answer to the big question: "How to multiply vectors?" Three geniuses, Hamilton, Grassmann and Clifford answered that question in 19. century! 

Once again: how?

1. Would You like multiplications almost like real numbers (associativity, distributivity ...)? (yes)

2. Do You expect vectors multiplication to be commutative? (no, cross product is not)

3. What one obtains multiplying two identical vectors? (real scalar)

4. Do vectors and real numbers commute? (yes)

Ok, multiplying parallel vectors one obtains real number, so, it is commutative:

if b = 3a, then ab = 3aa = 3a2 = ba.

What if vectors are orthogonal? Ok, let a and b to be orthogonal. Then, using Pythagoras theorem, 

(a+b)2 = a2 + b2 + ab + ba = a2 + b2 ,

it means that ab = - ba. Orthogonal vectors are anti-commutative. 

Now, it is all You need to deal with a great amount of mathematics: that is Hamilton, Grassmann and Clifford legacy. 

It is important to think really hard about this fundamental question: How to multiply vectors? It is the matter of a civilization  accomplishment. You can rethink, ignore given arguments, ok, but think again, if You accept given arguments, You must accept geometric algebra: it is the matter of mathematical logic! It means that You fallowing the mathematical arguments must get to a position of Hestenes, who worked out many problems. 

Let have two orthogonal vectors:

(ab)2 = abab = - abba = - ab2a = - a2b2 < 0 

Who needs the imaginary unit?

Best regards ...

Thank you dear Dr Miroslav Josipovic.

I have studied some of Hestenes's articles. I have derived the shrodinger and the Pauli quantum potential through the geometric algebra. However the Dirac quantum potential and Pauli and the schrodinger quantum potential has been derived by Hiley. But i derived them(except the Dirac case) through the Hestenes's geometric view. One of the interesting aspects of the geometric algebra is its real description of quantum mechanics. All phenomena take place in a real space but in quantum mechanics we have to represent them in a configuration space. I believe this is a mathematical tool and only a representation of what we do not understand it well yet.  If the geometric algebra  can be developed for many particle cases in relativistic regime, then it may describe interactions of particles like in quantum field theory at least for quantum electrodynamics. I think if hidden variables be determined then we do not need a wave function in a configuration space. When quantum mechanics has such fundamental problems , how does  we expect its composition with gravity which is described in real space give correct quantum gravity? ...

Thank you

Dear Rahmani,

You opened now some interesting questions, I would like to discuss some of it, but in a private correspondence (using messages).  Here just to conclude: we can solve some problems of physics in future, but many problems we have now just because we (I mean "we" as humans) do not have clear idea of vector multiplication. It means that geometric algebra is not  just another cute tricky thing: geometric algebra is the direct consequence of the non-commutative product of vectors (Clifford product). Physicist are still thinking in old categories, but I am sure that there is no choice.

If one analyzes 3D euclidean space in the new paradigm there are two striking things:

1. objects (scalars, pseudoscalars, vectors and bivectors) are part of incredible structure: complex numbers, spinors, rotors, hyper-complex numbers, projectors, Lie groups. Lie algebras, ..., all is here! Due to hyper-complex nature of objects there is no need for the fourth time dimension - formalism of special relativity is here, simple and easy to calculations. 

2. Electromagnetic field is just like a part of geometry of 3D space. It is amazing! For example, nilpotents of algebra have a just right form for electromagnetic field. EM field is like something born from 3D geometry, but we must to know how to multiply vectors. 

One of the fundamental problems of quantum gravity is time: in general relativity time is a part of spacetime, so, it is a vector, in quantum mechanics it is a parameter. Geometric algebra Cl3 gives special relativity naturally with time as a parameter and gives an elegant method to formulate general relativity (Jones, Baylis).  

Best ...

Faramarz Rahmani, thanks for the question! It is nice that there are researchers who are looking in the root of the problem... The colleagues' comments are interseting as well. However, all is circulating around the same notions and same principles. To go onward we have to start not from the consideration of physical systems at the atom size 10-10 m (which is the real of the quantum mechanical formalism), but from Planck's scale 10-35 m. For this a new mechanism of the motion of a particle is needed because the scale of 10-35 m is the minimum size of physical space, or in other words, this is the size of a cell of space that can be presented as a mathematical lattice of primary cells (in the mathematical language - a mathematical lattice of primary topological balls).

When a particle is mowing in the mathematical lattice of cells, it must interact with ongoing cells. So, as the result, there should appear a cloud of the spatial lattice excitations that accompany the particle. Then in real physical space the moving particle can be portrayed as follows: the core cell i surrounded with a cloud of excitations. All together this can be mapped to the formalism of conventional quantum mechanics operating in a phase space as the particle's wave \psi-function (with a singularity in the central part).

It is interesting, that the name for these excitations is obvious - "inertons", i.e. carriers of the force of inertia. Because inertia is a reaction on the side of space to the particle that just started to move.

In such a manner we can see that an abstract wave \psi-function determined in a phase space is the {particle + its inerton cloud} in real physical space...