Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Theory of low-energy strings?

  1. Mar 30, 2008 #1
    Could the string theory be interpreted as low-energy theory of elementary particles?
    As is known, the theory of strings has many interesting and important possibilities, but it cannot be verified because of the Planckian scale. Does arise the question: is it possible to construct the theory, similar to the existing theory of strings, but accessible for the experimental check?
    The present checked theory of the elementary particles – Standard Model - is of order of Compton wavelength scale. Thus, what theory we will obtain, if strings will be of order of Compton wavelength scale and of low energies, which correspond to the usual elementary particles? What we have in the case of the string theory?

    Basic postulates of the theory of the Planckian scale strings
    Enumerate the main of postulates, accepted in the simplest version of the string theory.
    1) In nature there are some one-dimensional objects, which are subordinated to relativistic wave equation and therefore called strings.
    2) Strings have certain size of the order of [tex]10^{-35} [/tex]m.
    3) Strings are characterized by the vibrational energy (which can be converted into the mass).
    4) The simplest strings are the open mass-free boson strings.
    (Obviously, this means that they must move in the empty space with the speed of light).
    5) In nature there are closed strings with different number of loops, which are formed by twirling and closing of the open strings.
    (But, let us note that in the theory of strings the reasons for the twirling of the open strings are not known; theory either postulates them or is limited to purely mathematical conversions. Let us note also that such strings cannot obviously move with the speed of light, since the mass of the twirled string remains concentrated in the specific place of space).
    6) Masses of the twirled strings are determined by the intensity of oscillations, and spins by a number of loops and by their motion.
    7) At least the part of such excited strings must correspond to known elementary particles. (since other particles (e.g. super-particles) cannot be fixed on the usual scale of lengths and energies, we will not speak about them).
    8) The theory of strings formally makes possible to construct the gravity equation of Einstein.

    What objects of the usual scale of lengths and energies we can compare to these strings?
    Last edited: Mar 30, 2008
  2. jcsd
  3. Mar 30, 2008 #2


    User Avatar
    Gold Member

    The objection is that the string scale, given by its tension, should be something around 1GeV - 1 Tev. So we should expect to detect form factors and compositeness in the electron.

    I suspect it could be possible to bypass this objection because the particles in the standard model are not the bosonic states but the fermionic ones; I havent see a serious calculation of the cross section for scattering of fermionic strings and it could be that its scattering size were a lot smaller than its intrinsic tension scale.

    Hey, do you have a proposal? tell us!
    Last edited: Mar 30, 2008
  4. Mar 30, 2008 #3
    particles in the standard model are not the bosonic states but the fermionic ones;

    How about pseudoscalar mesons?There are bosons..
    and phenomenon of 18 degrees my be the manifestation some strings.
    Proton is basic tone and mesons are overtones..
  5. Mar 31, 2008 #4
    I talk about the low-energy theory of strings, not about the Standard Model.

    Let us attempt to find first the low-energy bosonic strings, then we can look if can appear the fermionic states. So:

    What objects of the usual scale of lengths and energies we can compare to these strings?

    Open strings of the low-energy scale?
    What objects of the usual scale of lengths and energies we can compare to these strings? Is there a similar object in the microcosm? Let us begin from the simplest mass-free boson string.
    As is known, in nature there is only one object, which has zero mass and simultaneously it is a boson: this is a photon. But does have it other corresponding characteristics? Let us verify this.
    Since we do not know the structure of photon, we can make conclusions only on the basis of its mathematical characteristics.
    1) Is subordinated a photon to relativistic wave equation?
    In the framework of QED (Akhiezer and Berestetskii, p. 15) “as field equation, which describes quantum mechanical state of photons or photon system, is naturally to take the Maxwell equations. It is not difficult to show that this assumption together with relation is sufficient to build the theory of photon and its interaction with other particles”.
    Since the relativistic electromagnetic wave equation follows from the Maxwell equations, from above follows that a photon must also subordinated to the same wave equation, with difference that its frequency (energy) are quantized.
    Thus, photon is actually the mass-free boson, which is subordinated to wave equation.
    2) Following question: is it possible to consider a photon as one-dimensional object?
    One-dimensional object is characterized by one parameter of size, called length. What we do have in the case of photon?
    According to Planck and Einstein (Frauenfelder and Henley) monochromatic electromagnetic (EM) wave consists of the [tex]N[/tex] monoenergetic photons, each of which has zero mass, energy [tex]\varepsilon [/tex], momentum [tex]\vec {p}[/tex], and wavelength [tex]\lambda [/tex], whose values are one-to-one connected between them: [tex]\varepsilon =\hbar \omega ,\vec {p}=\hbar \vec {k}[/tex], [tex]\varepsilon =cp[/tex], ([tex]\vec {k}={\vec {k}^0} \mathord{\left/ {\vphantom {{\vec {k}^0} \mathchar'26\mkern-10mu\lambda }} \right. \kern-\nulldelimiterspace} \mathchar'26\mkern-10mu\lambda [/tex] is wave vector, [tex]\mathchar'26\mkern-10mu\lambda=\lambda \mathord{\left/{\vphantom{\lambda {2\pi }}} \right. \kern-\nulldelimiterspace} {2\pi }[/tex] is the shortened wavelength). The number of photons in EM wave is such, that their total energy is equal [tex]\varepsilon _{full} =N\varepsilon =N\hbar \omega [/tex].
    Photons are bosons and coherent photons are capable to be condensed in the EM wave (for example, in the form of laser beam), which has the specific frequency.
    Thus, since the photon characteristics are one-to-one connected with each other, in order to describe photon it suffices to know only its one parameter: wavelength.
    Now, it remained to prove that the region of space, in which was concluded the photon, is characterized by its wavelength.
    3) From a theoretical point of view the proof was given long years ago in the work of Landau and Peierls and confirmed recently in the works of other scientists (Cook, Inagaki and others). Let us consider briefly this proof, using the book of (Akhiezer and Berestetskii, 1969):

    The wave function of photon is here introduced as follows. The vectors of the EM field [itex]\vec {{\rm E}}[/itex] and [itex]\vec {{\rm H}}[/itex], as the solutions of the wave equation of the second order, which follow from the Maxwell equations, are considered as the classical wave functions [itex]\vec {\varepsilon }\left({\vec {r},t} \right)[/itex] and [itex]\vec {H}\left( {\vec {r},t} \right)[/itex].

    Representing the wave equation as multiplication of two equations for the advanced and retarded waves, we obtain two linear equations, which correspond to the wave vector [itex]\vec {f}_k [/itex] and is a certain generalization of vectors of EM field. The equation for this function is equivalent to the system of the Maxwell equations. For this reason it is possible to consider the Maxwell's equation as the equation of one photon (Gersten, 2001). The quantization of classical wave function is produced by means of the quantization of energy of this wave by the introduction of the relationship [itex]\varepsilon =\hbar \omega [/itex]. It turned out that in this case the function [itex]\vec {f}_k [/itex] could be interpreted as the quantum wave function of photon in the momentum space.

    But with the attempt to introduce the function of photon in the coordinate representation was revealed the insurmountable difficulty According to the analysis of Landau and Peierls (Landau and Peierls, 1930), and later of Cook (Cook, 1982a; 1982b) and Inagaki (Inagaki, 1994), the wave function of photon by its nature is nonlocal (see also the review (Bialynicki-Birula, 1994)). .

    Actually, after completing the inverse Fourier transformation of above function [itex]\vec {f}_k [/itex] we obtain:
    [itex]\frac{1}{\left( {2\pi } \right)^3}\int {\vec {f}_k e^{i\vec {k}\vec {r}}d^3k=\vec {f}\left( {\vec {r},t} \right)} [/itex].

    It seems that it is possible to determine [itex]\vec {f}\left( {\vec {r},t} \right)[/itex] as the wave function of photon in the coordinate representation. Actually, because of normalization condition for [itex]\vec {f}_k [/itex] the function [itex]\vec {f}\left( {\vec {r},t} \right)[/itex] will be also normalized by the usual method: [itex]\int {\left|{\vec {f}\left( {\vec {r},t} \right)} \right|} ^2d^3x=1[/itex]

    However, the value [itex]\left| {f(\vec {r},t)} \right|^2[/itex] will not have the sense of the probability density distribution to find the photon at the given point of space. Actually, the presence of photon can be established only by its interaction with the charges.

    This interaction is determined by the values of the EM field vectors [itex]\vec {{\rm E}}[/itex] and [itex]\vec {{\rm H}}[/itex] at the given point, but these fields are not determined by the value of the wave function [itex]\vec {f}\left( {\vec {r},t} \right)[/itex] at the same point, and they are defined by its values in entire space.

    In fact, the component of the Fourier field vectors, expressed by[itex]f_k [/itex], contain the factor [itex]\sqrt k [/itex]. Formally this can be written down in the form
    \[\vec {\varepsilon }\left( {\vec {r},t} \right)=\sqrt[4]{-\Delta }\vec {f}\left( {\vec {r},t} \right)\]
    where [itex]\Delta [/itex] is the Laplace operator. But [itex]\sqrt[4]{-\Delta }[/itex] is integral operator, and therefore the relationship between [itex]\vec {\varepsilon }\left( {\vec {r},t} \right)[/itex] and [itex]\vec {f}\left( {\vec {r},t} \right)[/itex] is not local, but integral. In other words, [itex]\vec {f}(\vec {r},t)[/itex] is not determined by field value [itex]\vec {{\rm E}}(\vec {r},t)[/itex] at the same point, but it depends on field distribution in a certain region, whose size is the order of wavelength.
    This means that localization of photon in the smaller region is impossible and, therefore, the concept of the probability density distribution to find the photon at the fixed point of space does not have a sense.
    This conclusion of theory is confirmed by experiment, since all measurements with the use of EM waves or photons (interference, diffraction and so forth) can be carried out to the region, not smaller as wavelength.
    Thus, one of the fundamental particle of EM field (photon) can be described as one-dimensional relativistic string of one wavelength size, whose value corresponds to its energy according to the Planck formula.

    Thus formally we can name a photon as open quantum electromagnetic (QEM) string

    Now, what about a close strings?{/B]
  6. Mar 31, 2008 #5
    Sorry, I forgot the biblιography

    Akhiezer, A.I. and Berestetskiy, V.B. (1965). Quantum electrodynamics.
    Bialynicki-Birula, Iwo (1994) On the wave function of the photon. Acta physica polonica, 86, 97-116),
    Cook, R.J. (1982a). Photon dynamics. A25, 2164
    Cook, R.J. (1982b). Lorentz covariance of photon dynamics. A26, 2754
    Frauenfelder, H. and Henley, E.M. (1974). Subatomic physics, Prentice-Hall, Inc., New Jersey,.
    Gersten, A. (2001) Maxwell equation - the one-photon quantum equation. Found. of Phys., Vol.31, No. 8, August).
    Inagaki, T. (1994). Quantum-mechanical approach to a free photon. Phys. Rev. A49, 2839.
    Landau, L.D and Peierls, R. (1930). Quantenelekrtodynamik in konfigurationsraum. Zs. F. Phys., 62, 188.
    Last edited: Mar 31, 2008
  7. Mar 31, 2008 #6


    User Avatar
    Gold Member

    Well you did not gave any hint, so I answered from my own guess: QCD strings for the bosonic part, Standard Model for the fermionic part. This is motivated because physically we can classify the QCD strings into 6 charged +1, 6x3 coloured -1/3, 6x3 coloured +2/3, 13 neutral, plus the antiparticles of the first three groups, plus (3+3)x3 coloured \pm 4/3. On the other hand, the Standard Model fermions are 6 charged +1, 6x3 coloured -1/3, 6x3 coloured +2/3, 12 neutral, plus the antiparticles of the first three groups. So it makes sense to consider both as fermionic and bosonic counterparts of a superstring theory.

    On other hand, what energy is low energy? 1 TeV as in the LHC? 1 GeV as QCD? Or 0.001 eV as neutrino mass? Note also that a pair of these scales determine a high scale, consider seesaws

    (1 GeV)^2 / 0.001 eV = 10^21 eV = 10^12 GeV
    (1 TeV)^2 / 0.001 eV= 10^27 eV = 10^18 GeV

    and compare with Planck mass = 10^18-10^19 GeV or GUT or sGUT masses, around 10^15-10^16 GeV.

    It is an interesting idea, but does it really match with the ideas and formalism of string theory, or is it just using "string" as a name?
  8. Mar 31, 2008 #7
    This is very correct conclusion.

    And this is very good question. I think that the answer to it you will be able to formulate themselves after we will obtain the equations of closed strings.
  9. Mar 31, 2008 #8
    Closed low-energy strings?

    What we can say generally about QEM closed strings (QEM - particles) ?
    Now we must show that the open QEM-string can form the closed strings with different number of loops, which would possess the characteristics of known particles. Let us try to begin from the simplest closed strings, which correspond to one simplest loop - a ring.
    But, first of all, let us answer the question, to which the theory of the strings does not give the answer: what reason does force a string to twirl into the loop?
    It is not difficult to recall that the trajectory of the motion of electromagnetic wave can be bent in the strong field (for example, in the field of atomic nucleus, neutron star and the like) or in the medium with the variable refractive index. Thus, we have a base to assert that the open QEM-string also can change the trajectory of its motion within the strong electromagnetic field and move along the curvilinear trajectory, such as ring.

    It is necessary to note here that apparently the “linear” Maxwell fields of open string will not be Maxwellian after the twirling. These will be some non-linear fields, described by some non-linear equations, which are not the fields and equations of classical theory.

    There is also one additional question, which is very important for the verification of the assumed theory. A question is how from the nonlocal string with size of the order of Compton wavelength can arise local, i.e., point fundamental particles - leptons and quarks.

    The theory of strings solves this problem, relying on its non-verifiability. It asserts that the point elementary particles in reality have a size of Planckian scale, but since we did not experimentally reach this accuracy, we cannot assert, that theory is incorrect.

    In our case to use this trick it is impossibly, since Compton lengths is long time ago accessible for our experiments. At the same time among the theorists the steadfast persuasion exists that the fundamental particles - lepton and quarks – are pointness. This persuation is based on the theoretic arguments as well as on the correct experimental results. Leaving for the future the analysis of these questions, we confine here oneself to the following goal: if we want to prove the possibility of production of fundamental particles (leptons and quarks) from the QEM-strings, we must derive the elementary particle equations, which do not contain particle size.

    In other words a question is to show that the equations of motion of the twirled QEM-strings are absolutely identical to the equations of elementary particles (as the Klein-Gordon, Dirac, Yang-Mills, Procá and other equations), which, as it is known, don’t contain the terms with the sizes of particles.

    Try to show further that this possibility exists and don’t demands no trick.
    Last edited: Mar 31, 2008
  10. Mar 31, 2008 #9
    Αn open string transformation into a closed string

    So, let us suppose that under some conditions (e.g. strong EM field) a “linear” open QEM-string is able to twirl and move along the closed curvilinear trajectory. For short we will name this transformation as “twirl transfomation” or “ twirling”. Try to translate below this supposition on the mathematical language, beginning from the open string description.
  11. Mar 31, 2008 #10


    User Avatar
    Gold Member

    I hope you are not claimed we have already obtained the equations of open strings... I havent see any action for a worlsheet in the above discussion. Did I missed it?

    Also, I haven't see in your QED argument any energy scale explaining why do you call it "low energy"; the only scale in your argument is the wavelenght of the photon, there is not limit for it.
  12. Mar 31, 2008 #11
    Open QEM-string equation in the matrix form

    Let us consider the plane electromagnetic (EM) wave moving, for example, on [tex]y[/tex]- axis. As is known the electric and magnetic fields can be written in the complex form as:
    (eq1) [tex]
    \left\{ {\begin{array}{l}
    \vec {{\rm E}}=\vec {{\rm E}}_o e^{-i\left( {\omega t\pm ky} \right)}, \\
    \vec {{\rm H}}=\vec {{\rm H}}_o e^{-i\left( {\omega t\pm ky} \right)}, \\
    \end{array}} \right.

    The electromagnetic wave of any direction has two plane polarizations and contains only four field vectors; for example, in the case of [tex]y[/tex]-direction we have:

    (eq2) [tex]
    \vec {\Phi }(y)=\left\{ {{\rm E}_x ,{\rm E}_z ,{\rm H}_x ,{\rm H}_z }

    and [tex]{\rm E}_y ={\rm H}_y =0[/tex] for all transformations.

    The EM wave equation has the following view [6]:

    (eq3) [tex]
    \left( {\frac{\partial ^2}{\partial t^2}-c^2\vec {\nabla }^2} \right) \vec
    {\Phi }(y)=0,

    where [tex]\vec {\Phi }(y)[/tex] is any of the above electromagnetic wave fields (eq2).
    In other words this equation represents four equations: one for each wave function of the electromagnetic field.
    We can also write this equation in the following operator form:

    (eq4) [tex]
    \left( {\hat {\varepsilon }^2-c^2\hat {\vec {p}}^2} \right)\Phi (y)=0,

    where [tex]\hat {\varepsilon }=i\hbar \frac{\partial }{\partial t}, \quad \hat {\vec
    {p}}=-i\hbar \vec {\nabla }[/tex] are the operators of the energy and momentum correspondingly and [tex]\Phi [/tex] is some matrix, which consists four components of [tex]\vec {\Phi }(y)[/tex].
    Taking into account that

    [tex]\left( {\hat {\alpha }_o \hat {\varepsilon }}\right)^2=\hat {\varepsilon }^2, \quad \left( {\hat {\vec {\alpha }}\hat {\vec {p}}} \right)^2=\hat {\vec {p}}^2[/tex],

    where [7,8] [tex]\hat {\alpha }_0 ; \quad \hat {\vec {\alpha }}; \quad\hat {\beta }\equiv \hat {\alpha }_4 [/tex] are Dirac's matrices and [tex]\hat {\sigma}_0 [/tex],[tex]\hat {\vec {\sigma }}[/tex] are Pauli matrices, the equation (eq4) can also be represented in the matrix form of the Klein-Gordon-like equation without mass:

    (eq5) [tex]
    \left[ {\left( {\hat {\alpha }_o \hat {\varepsilon }} \right)^2-c^2\left(
    {\hat {\vec {\alpha }}\hat {\vec {p}}} \right)^2} \right]\Phi =0

    Taking into account that in case of photon [tex]\omega =\frac{\varepsilon }{\hbar }[/tex] and [tex]k=\frac{p}{\hbar }[/tex], from (eq5), using (eq1), we obtain [tex]\varepsilon =cp[/tex], as for a photon. Therefore we can consider [tex]\Phi [/tex] as wave function of the equation (eq5}) both as EM wave and as a photon.

    Factorizing (eq5) and multiplying it from left on the Hermitian-conjugate function [tex]\Phi ^+[/tex] we get:

    (eq6) [tex]
    \Phi ^+\left( {\hat {\alpha }_o \hat {\varepsilon }-c\hat {\vec {\alpha
    }}\hat {\vec {p}}} \right) \left( {\hat {\alpha }_o \hat {\varepsilon
    }+c\hat {\vec {\alpha }}\hat {\vec {p}}} \right)\Phi =0,

    The equation (eq6) may be disintegrated on two Dirac equations without mass:

    (eq7) [tex]
    {\Phi ^+\left( {\hat {\alpha }_o \hat {\varepsilon }-c\hat {\vec {\alpha
    }}\hat {\vec {p}}} \right)=0,} \hfill \\
    {\left( {\hat {\alpha }_o \hat {\varepsilon }+c\hat {\vec {\alpha }}\hat
    {\vec {p}}} \right)\Phi =0,} \hfill \\
    \end{array} }

    each of which we will further conditionally name the linear semi-photon equations.

    It is not difficult to show that only in the case when we choose the [tex]\Phi [/tex]-matrix in the following form:

    (eq8) [tex]
    \Phi =\left( {{\begin{array}{*{20}c}
    {{\rm E}_x } \hfill \\
    {{\rm E}_z } \hfill \\
    {i{\rm H}_x } \hfill \\
    {i{\rm H}_z } \hfill \\
    \end{array} }} \right),
    \Phi ^+=\left( {{\begin{array}{*{20}c}
    {{\rm E}_x } \hfill & {{\rm E}_z } \hfill & {-i{\rm H}_x } \hfill & {-i{\rm
    H}_z } \hfill \\
    \end{array} }} \right),

    the (eq7) are the right Maxwell equations of the electromagnetic waves: retarded and advanced. Actually using (eq8) and putting in (eq7) we obtain:

    (eq9) [tex]
    {\left\{ {\begin{array}{l}
    \frac{1}{c}\frac{\partial {\rm E}_x }{\partial t}-\frac{\partial {\rm H}_z
    }{\partial y}=0 \\
    \frac{1}{c}\frac{\partial {\rm H}_z }{\partial t}-\frac{\partial {\rm E}_x
    }{\partial y}=0 \\
    \frac{1}{c}\frac{\partial {\rm E}_z }{\partial t}+\frac{\partial {\rm H}_x
    }{\partial y}=0 \\
    \frac{1}{c}\frac{\partial {\rm H}_x }{\partial t}+\frac{\partial {\rm E}_z
    }{\partial y}=0 \\
    \end{array}} \right.,} \hfill & {\left\{ {\begin{array}{l}
    \frac{1}{c}\frac{\partial {\rm E}_x }{\partial t}+\frac{\partial {\rm H}_z
    }{\partial y}=0 \\
    \frac{1}{c}\frac{\partial {\rm H}_z }{\partial t}+\frac{\partial {\rm E}_x
    }{\partial y}=0 \\
    \frac{1}{c}\frac{\partial {\rm E}_z }{\partial t}-\frac{\partial {\rm H}_x
    }{\partial y}=0 \\
    \frac{1}{c}\frac{\partial {\rm H}_x }{\partial t}-\frac{\partial {\rm E}_z
    }{\partial y}=0 \\
    \end{array}} \right.} \hfill \\
    \end{array} },

    (for waves of any other direction the same results can be obtained by the cyclic transposition of the indexes and by the canonical transformation of matrices and wave functions [9].
  13. Apr 1, 2008 #12
    No, you missed nothing

    But, as it is known, in the simplest case the action for the string theory can be reduced to action for the classical relativistic wave equation. Compare, e.g., Polyakov’s action for the boson string and action for the wave motion.

    Our purpose is to find the dynamic equations of motion of fields (particles). Characteristic functions for the relativistic wave equation (Lagrangian, Hamiltonian, action) are familiar. There is no sense to begin from them, when it is possible to begin from the dynamic equations. Above we obtained them as equations of QFT.

    Now we can consider the twirl transformation.
  14. Apr 1, 2008 #13
    About the twirl transformation of fields

    The transformation of the “linear” QEM-string to the curvilinear, briefly -- ``twirl transformation'') can be conditionally represented as following expression:

    (eq10) [tex]
    \hat {R}\Phi \to \Psi ,

    where [tex]\hat {R}[/tex] is the operator of some transformation of QEM-string from linear to curvilinear trajectory, the [tex]\Phi [/tex] is the open QEM-string wave function, defined by matrix (eq8), which satisfies the linear equations (eq5) and (eq7), and [tex]\Psi [/tex] is a closed QEM-string wave function, namely:

    (eq11) [tex]
    \Psi ^=\left( {{\begin{array}{*{20}c}
    {{\rm E}'_x } \hfill \\ {{\rm E}'_z } \hfill \\ {i{\rm H}'_x }
    \hfill \\ {i{\rm H}'_z } \hfill \\\end{array} }} \right),

    which appears after non-linear transformation, where [tex]\left( {{\begin{array}{*{20}c} {{\rm E}'_x } \hfill & {{\rm E}'_z } \hfill & {-i{\rm H}'_x } \hfill & {-i{\rm H}'_z } \hfill \\\end{array} }} \right)[/tex] are electromagnetic fields after twirl transformation. Note that this is not the linear Maxwelian quantized EM field, but some new non-linear EM quantized field, which doesn’t exists in classical physics.
    Note that mathematically this transformation is equivalent to the vector transition from flat space to the curvilinear space, which is described by Riemann geometry [10]. As it is known, the description of vector transition from linear to curvilinear trajectory is also fully described by usual differential geometry.
    Try also to understand the link of twirl transformation with known transformations of modern physics.
  15. Apr 1, 2008 #14
    The twirl transformation description in Rieman geometry

    We can consider the equations (7) as Dirac's equation without mass and simultaneously as the vector equation (9) of electromagnetic wave fields. In the usual form this equation has the following view: [tex]\left( {\hat {\alpha }_o \frac{\partial }{\partial _ t}-c\hat {\vec {\alpha }}\vec {\nabla }} \right)_ \Phi =0[/tex], and in 4-vector form: [tex]\hat {\alpha }_\mu \partial _\mu
    \Phi =0[/tex], where [tex]\hat {\alpha }_\mu =\left\{ {\hat {\alpha }_0 ,\hat {\vec {\alpha }}} \right\}[/tex], [tex]\partial _\mu \equiv \partial /\partial _ x_\mu [/tex]
    (here [tex]\mu =0, 1, 2, 3[/tex] are summation indexes, [tex]x_\mu [/tex] are the coordinates of 4-space).

    The generalization of this equation on the curvilinear (Riemann) geometry is connected with the parallel transport of the vector in the curvilinear space
    (Fock, 1929a,b; Van der Waerden, 1929; Goenner, 2004). It was shown that for this generalization it is enough to replace the usual derivative [tex]\partial _\mu \equiv \partial /\partial _ x_\mu [/tex] with the covariant derivative: [tex]D_\mu =\partial _\mu +\Gamma _\mu [/tex], where [tex]\Gamma _\mu [/tex] is the analogue of Christoffel's symbols in the case of the plane motion, called Ricci symbols (or Ricci connection coefficients).

    When a vector transports along the straight line, all the symbols [tex]\Gamma _\mu =0[/tex], and we have a usual derivative. But if a vector moves along the curvilinear trajectory, not all [tex]\Gamma _\mu [/tex] are equal to zero and a supplementary term appears.

    Typically, the last one is not the derivative, but it is equal to the product of the vector itself with some coefficient [tex]\Gamma _\mu [/tex], which is an increment. In the theory it shown that [tex]\hat {\alpha }_\mu \Gamma _\mu =\hat {\alpha }_0 p_0 +\hat {\alpha }_i p_i [/tex], where [tex]\left\{ {p_0 ,p_i } \right\}[/tex] are the real values with dimension of the 4-vector energy-momentum. Therefore, it is logical to identify [tex]\Gamma _\mu _{
    }[/tex] with 4-vector of energy-momentum of the QEM-string fields: [tex]\hat {\alpha }_\mu \Gamma _\mu =\hat {\alpha }_0 \varepsilon _p +\vec {\hat {\alpha }}_ \vec {p}_p [/tex], where [tex]\varepsilon _p [/tex] and [tex]p_p [/tex] is the QEM-string energy and momentum (not the operators).

    Taking into account that according to energy conservation law [tex]\hat {\alpha }_0 \varepsilon _p +\vec {\hat {\alpha }} \vec {p}_p =\pm \hat {\beta }
    m_p c^2[/tex], it is not difficult to see that the supplementary term contains a closed QEM-string mass.
  16. Apr 1, 2008 #15


    User Avatar
    Gold Member

    It is not known by me. Would you like to stop your equation dumping and illustrate this point? Also, if it happenst in the simplest case, can I assume you are speaking of the simplest case here?

    I still believe you are using string theory as a name, not as a theory.
  17. Apr 2, 2008 #16
    I will attempt briefly (without details) to explain what I mean.
    First of all, general remarks: as it is known, elementary particles are also waves and all equations of QFT (i.e. of Standard Model) are wave equations. This cannot be abolished by any new theory, since SM is very well checked experimentally.
    This fact alone indicates that the theory of strings, if it is intend for describing of elementary particles, must reduce to the wave equations, or, in other words, must have Lagrangian and actions of the wave equations.

    Now more concrete about equations, Lagrangians and actions:

    In the simplest case of real, relativistic-invariant field with one component, the equation of motion is written as:
    \frac{1}{c^2}\mathop \psi \limits^{\ast \ast } +\Delta \psi =0
    The following Lorentz-invariant Lagrangian corresponds to it:
    \bar {L}=-\frac{1}{2}c^2\sum\limits_\nu {\frac{\partial \psi }{\partial
    x_\nu }} \frac{\partial \psi }{\partial x_\nu }\equiv
    -\frac{1}{2}c^2\partial _\nu \psi \partial ^\nu \psi
    and also the action:
    I=-\frac{1}{2}c^2\int\limits_t {dt} \int\limits_V {\partial _\nu \psi
    \partial ^\nu \psi _ dx}
    Now, what we do have in the theory of strings? First of all let us, note that here is examined the ormalism of multidimensional curvilinear space and therefore the theory formulas are complicated due to different of factors and coefficients (Christoffel etc).

    As initial Lagrangian the relativistic Lagrangian of point particle motion is used here. As generalization of the last we obtain the Nambu-Goto action:
    S(x)=-\frac{1}{2}T\int {d\sigma } \int {d\tau _ } \sqrt {\dot
    {x}^2x^{'2}-(\dot {x}\cdot x^')^2}
    The square root in the Nambu-Goto action make the quantum treatment complicated. So we introduce an equivalent (by Polakov) action, which does not have the square root:
    S(x,\gamma )=-\frac{1}{2}T\int {d\sigma } \int d \tau \sqrt {-\gamma }
    \gamma ^{ab}\partial _a x^\mu \partial _b x^\nu \eta _{\mu \nu }
    To pass to the real elementary particles in the framework of SM we must consider the functions [tex]x^\mu[/tex] as wave functions (taking into account the vibration of strings, etc). In this case we see that this action is similar to above action of the wave equations.

    As you know, any theory can be written down and formulated by many different methods (for example, in the classical mechanics it is possible to use the approach of Newton, Lagrange, Jacobi, Hamilton, etc. In quantum mechanics it is possible to use the matrix mechanics of Heisenberg, wave mechanics of Schroedinger, integrals along the paths of Feynman, etc)

    Furthermore, same equations can be written down in the form of the expressions of the different level of abstraction (about this see, e.g. the Feynman Nobel lecture).

    For example, the equations of electrodynamics can be written down in the form of eight scalar equations, in the form of four vector equation, in the form of two tensor equations, and also in the form of the quaternion and octanion equations (and maybe, in other forms). Furthermore it is possible to write down them in 11 different orthogonal systems, and also in the curvilinear space of many dimensions, etc

    But all these approaches (as Feynman remarked in his lectures, volume “Electrodynamics”) give the same results.

    Moreover, in order to calculate something in the electrodynamics, it is necessary to turn to eight scalar equations. In other words, all methods of description, enumerated above, are actually useless.

    Now I will answer your remark (“I still believe you are using string theory as a name, not as a theory”):
    it is possible to say that the method to set out the theory of Standard Model by means of the strings is one of the methods of describing of this theory. In other words: this is the string interpretation of SM.

    But in my approach another is important: it occurs that we can explain many things, which in the abstract theories make only mathematical sense (for example, in this interpretation the strings have compound field structure; they can be twirled, broken in two other strings, superposed one on another; and many others.)

    Maybe, namely this is the only destination of string theory, and all the other is only the formal abstract construction? I don’t know.
  18. Apr 2, 2008 #17
    Consider now

    The twirl transformation description in differential geometry

    Let the plane-polarized wave, which has the field vectors [tex](E_x ,H_z )[/tex], be twirled with some radius [tex]r_p [/tex] in the plane [tex](X',O',Y')[/tex] of a fixed co-ordinate system [tex](X',Y',Z',O')[/tex] so that [tex]E_x [/tex] is parallel to the plane [tex](X',O',Y')[/tex] and [tex]H_z[/tex] is perpendicular to it (fig 1).

    According to Maxwell [6] the displacement current in the equation (eq9) is defined by the equation:

    (eq12) [tex]
    j_{dis} =\frac{1}{4\pi }\frac{\partial \vec {E}}{\partial t},

    The above electrical field vector [tex]\vec {E}[/tex], which moves along the curvilinear trajectory (let it have direction from the center), can be written in the form:

    (eq13) [tex]
    \vec {E}=-E\cdot \vec {n},

    where [tex]E=\left| {\vec {E}} \right|[/tex], and [tex]\vec {n}[/tex] is the normal unit-vector of the curve (having direction to the center). The derivative of [tex]\vec {E}[/tex] can be represented as:

    (eq14) [tex]
    \frac{\partial \vec {E}}{\partial t}=-\frac{\partial E}{\partial
    t}\vec {n}-E\frac{\partial \vec {n}}{\partial t},

    Here the first term has the same direction as [tex]\vec {E}[/tex]. The existence of the second term shows that the additional displacement current appears at the twirling of the wave. It is not difficult to show that it has direction, tangential to the ring:

    (eq15) [tex]
    \frac{\partial \vec {n}}{\partial t}=-\upsilon _p \kappa \vec {\tau },

    where [tex]\vec {\tau }[/tex] is the tangential unit-vector, [tex]\upsilon _p \equiv c[/tex] is the electromagnetic wave velocity, [tex]\kappa =\frac{1}{r_p }[/tex] is the curvature of the trajectory and [tex]r_p [/tex] is some curvature radius. Thus, the displacement current of the plane wave, moving along the ring, can be written in the form:

    (eq16) [tex]
    \vec {j}_{dis} =-\frac{1}{4\pi }\frac{\partial E}{\partial t}\vec
    {n}+\frac{1}{4\pi }\omega _p E\cdot \vec {\tau },

    where [tex]\omega _p =\frac{m_p c^2}{\hbar }=\frac{\upsilon _p }{r_p }\equiv c\kappa [/tex] we name the curvature angular velocity, [tex]\varepsilon _p =m_p c^2[/tex] is photon energy, [tex]m_p [/tex] is some mass, corresponding to the energy [tex]\varepsilon _p [/tex], [tex]\vec {j}_n =\frac{1}{4\pi }\frac{\partial E}{\partial t}\vec {n}[/tex] and [tex]\vec {j}_\tau =\frac{\omega _p }{4\pi }E\cdot \vec {\tau }[/tex] are the normal and tangent components of the current of the twirled electromagnetic wave, correspondingly. Thus:

    (eq17) [tex]
    \vec {j}_{dis} =\vec {j}_n +\vec {j}_\tau ,

    The currents [tex]\vec {j}_n [/tex] and [tex]\vec {j}_\tau [/tex] are always mutually perpendicular, so that we can write them in the complex form:

    (eq18) [tex]
    j_{dis} =j_n +ij_\tau ,

    where [tex]j_\tau =\frac{\omega _p }{4\pi }E[/tex]. Thus the tangent current appearance cause the appearance of imaginary unit in QM theory. From the above we can also assume that the appearance of imaginary unit in the quantum mechanics is tied with the tangent current appearance.
  19. Apr 2, 2008 #18


    User Avatar
    Gold Member

    Justo to clarify: what we all see here is that the action in string theory is similar to the action of wave equations in 1+1 dimensions. Ok, if this is deep or it is not, now the rest of the readers can consider by themselves.

    Really I am sorry, because the title of this thread was more promising.
  20. Apr 3, 2008 #19
    Not only. We see here that the strings are not abstract mathematical objects, but physical field elements of nature.

    I hope that the thread gives further more than title promises.
  21. Apr 3, 2008 #20
    Physical sense of twirl transformation

    Let’s now revert to the question, what is the transformation, which we produced. In QFT and GTR [11] this transformation is known as the transformation of the field, which generates the compensate fields. It was shown also, that this transformation is equivalent to the gauge transformation, which generates the gauge fields. It is also known [11] that gauge transformations are the transformations of the internal symmetry of particles, as it is also has place in our case .
    In other words, the Ricci coefficients (or in the general case of the 3-D curvilinear motion, Christoffel's coefficients) are the compensating fields or gauge fields in the QFT. This is not difficult to show, if we write down energy and momentum of the intrinsic field of the closed string through the 4-potentials [tex]\hat {\alpha }_\mu \Gamma _\mu =\hat {\alpha }_0 \varepsilon _p +\vec {\hat {\alpha }}\vec {p}_p =\alpha _\mu A^\mu [/tex], where[tex]A_\mu [/tex] is 4-potential of EM field, which is in the QFT the gauge fields.
    It is not of difficult to see also that the tangent current [tex]j_\tau [/tex] corresponds to the Ricci connection coefficients (symbols) [tex]\Gamma _\mu [/tex]
    Thus, the transformations of the internal rotation of QEM-strings are mathematically identical to gauge transformations. In this connection let us remind that the Pauli matrices as well as the photon matrices (the generators of SU(2) and SU(3) groups) are the space rotation operators - in 2D and 3D space correspondingly [11].

    Thus, further we must show how due to twirling the open massless boson string is transformed in the closed massive boson string .
    Last edited: Apr 3, 2008
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?