Blogs of Professor(Dr.) Pranab Kumar Bhattacharyya MD(calcutta.Univ) Pathology; : July 2022

What every person must know about the string theories to evolve all super clusters, galaxies ,stars, planets , planetesimals in the Universe

Authors are -:

Rupak Bhattacharya ,BSc (Calcutta University), MSc (Jadavpur University) ,of residence 7/51 Purbapalli, Post office - Sodepur, District 24 parganas ( North) Kolkata -110, West Bengal, India,

Freelancer Theoretical physicist

( He has No institutional affiliation )

** Professor Dr Pranab Kumar Bhattacharya MD (University of Calcutta) ,FICPath, WBMES (Retired),

Ex Retired Professor and Head of Pathology from West Bengal Medical Education Services cadre of Department of Health and Family Welfare of Govt of West Bengal , India

( He Has Presently no Institutional affiliation)

**Upasana Bhattacharya,

student of 5th semester B.DES ( Fashion Design course ) , School of Design , UPES University ,Bidhouli , Derhadoon, Uttarakhand, India ,

***Ritwik Bhattacharya, B. Com

( Calcutta University)

( He has No institutional affiliation )

****Aiyshi Mukherjee ,

BSC Biotechnology (honors) 2nd yr student of Kalyani University, West Bengal

***Rupsha Bhattacharya BA (honors) 2nd yr student of Journalism & Mass communications of West Bengal State University, Barasat , North 24 Parganas West Bengal

****Debasish Mukherjee BSc (Calcutta University)

(He Has No institutional affiliation )

****Dalia Mukherjee BA honors (Calcutta university)

(She Has No institutional affiliation )

***Hindol Banerjee BA (honors) West Bengal State University, Barasat ,north 24 parganas, West Bengal, India

*** Of residence 7/51 Purbapalli , Post office --Sodepur, D-: 24 parganas

( north) ,Kolkata -110 ,West Bengal, India

(No institutional affiliation)

**** Of residence Swamiji Nagar, South Habra, District 24 Parganas ( north ),West Bengal ,India

(No institutional affiliation)

Acknowledgement -:

All authors of this article gratefully acknowledge contributions of following persons to bring them up and educate them about the" Origin and fate of this Universe, multi Universe theory, possibility of Extraterrestrial intelligent civilization, panspermia theory and exoplanets" like our parents late Mr. Bholanath Bhattacharya (1926-2009) B.Com ( honors in accountancy; University of Calcutta) ; FCA ( intermediate );SAS and late Mrs. Bani Bhattacharya (1935-2006) of their residence at 7/51 Purbapalli, Post Office -: Sodepur, District 24 Parganas (North ) ; Kolkata -110, West Bengal ,India and also to bellow mentioned uncles and aunts late Mr. Ajit Kumar Chakraborty, late Mrs Sudharani Chakraborty, late Mr Abani Kumar Chakraborty , late Mrs Rebeka Chakraborty , late Dr Asit Kumar Chakraborty, Mr. Binay Chakraborty, Mrs Aparna Chakraborty ,Prof .Dr . Monoj Bhattacharya, late Prof. Dr. Krishna Bhattacharya , late Mr. Nakul Chandra Bhattacharya and specially to Late Mrs Sialabala Chakrabortyn of Mahajati Nagar Agarpara, North 24 Parganas, West Bengal, India

Corresponding author -: Professor Dr Pranab Kumar Bhattacharya MD ( University of Calcutta ); FICPath; WBMES (Retired)

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Abstract-:

The basic things of string theory say that we live in an accelerating and ever expanding universe today. String theory also supports the inflation theory of the Big Bang, where a period of rapid expansion happened in the history of the early Universe. Most of the theories in string theory are focused on understanding a theory of broken Supersymmetry. In String theory, De -Sitter space can arise only when Supersymmetry is broken. Breaking supersymmetry in string theory required us to come face to face with problems to maintain stabilisation of the universe. In String theory, vacuum ( voids) was created along with N≥ 2 supersymmetry; there may be flat universes or modules. The energy as we go along three directions in space time ( as it is in scalar field theory) ,there are rather many flat directions or dimensions in string theory . In field theory, the space is remaining constant and in fact vanishes immediately. But there are 100 flat directions in compactification in string theory .The flat directions or dimensions are however very bad in Big Bang standard model cosmology. Flat directions cause however many problems in the standard Big Bang model.

The big question to us is that , does String theory allows the De- sitter universe ,- a Universe with negative cosmological constant to anti De- sitter space and inflation theory of Big Bang? It is known that De -sitter space can only arise if and when Supersymmetry is broken. In string theory N ≥2 Supersymmetry there are many flat directions or dimensions .The energy as we go along then these spaces is a constant and intact vanishes identically and these directions are bad news from part of cosmology . Cosmologically flat directions also ruin successful prediction of nucleosynthesis later . However a new class of string compactification called "flux compactification" are there. In this compactification besides curling up the extra dimensions as presumed in string theory , small size fluxes are also turned on along the compactified directions. The fluxes includes higher form generalization of magnetic flux in electromagnetism turning them on charges, the potential in modules space, so that new minima arise in regions or feild space where the potential can be calculated there with control. The value of cosmological constant in this minima can be calculated with a positive value given in De- sitter universe. This theory of Rupak Bhattacharya, Ritwick Bhattacharya, Prof Dr Pranab Kumar Bhattacharya etal (here are authors) can allow De- sitter universe and inflation at Big Bang in "M theory" and why then this universe started from a Universe that went to re collapsed and faced a Big crunch ie Big Bounce, within a Planck time of Big Bang ( Reference no 1- Rupak Bhattacharya ,Dr Pranab Kumar Bhattacharya upasana Bhattacharya Ritwick Bhattacharya etal "What if the Big Bang was not the beginning point of our observable universe" Research and Reviews : Journal of Space Science and Technology of STM journal ;vol 10 ;issue no 3 ; 2021; page 1-7)

String theory is the theory of everything? Whether string theory can live up to its claim of being the theory of everything and whether it will ever provide a satisfactory falsifiable prediction, as such remain a debated question before these present authors. String theory was first born in the 1960s. Here strings interactions demands the fundamental forces of the universe a) Gravity b) Electromagnetism c) Weak nuclear forces d) Strong nuclear forces as we will discuss in the text of this article, Fundamental particles scattering experiments in high energy particle accelerators like Large Hadron colliders ( LHC) had revealed a stunting proliferation of all resultant fundamental particles. Particles after particles like axions, excitators, spinors, resonance, quarks,neutrinos, W Bosons ,winos , all Fermions, R particles, at particular energies aimnantinig for over growing 200 of particles. But none of these particles including even the latest one of Nobel laureate Peter Higgs " Higgs particles" has yet proved to have resting zero mass in Gev/c*2 temperature in Grand Unified Theories ( GUTs) and thus can not be the most elementary particles in the universe with zero rest mass . Moreover, an over growing near zero mass of particles in motion at speed of light, can't be all elementary particles and thus a zero resting mass particle will be when discovered only then the mystery of the universe will be solved. We have to search for the resting zero mass particles in the colliders . According to the authors of this article, string theory had several flaws , most serious that the mathematical consistency of string theory at first told us that a string must have 26 special dimensions, rather than when our human eyes can see only three dimensions ( maximum four dimensions when space time is considered another dimension or directions). Later a rescue attempt replaced the string with new fermionic particle variety with infinitesimal particle's spins attached to tubular force that could braught dimensions from 26 to minimum 10. Thus in the 1970s QCD theory came.The Quantum field Gauge theory in QED of strong interaction resulted in a field blow to understand nuclear physics in terms of Superstring theory , where string interactions became weak at very short distances. But for the string, couplings that occur over large distances know a lattice QCD because it is successful to explain strings. We authors wonder if the strings theory stands as theory of everything , then despite all our efforts we still cannot explain why there are no free quarks in Nature ?

Key words

Cosmic String , Hadrons,quarks, anti quarks, Big Bang standard Model, GUT , super heavy strings, Gluons Minkowski space, oscillation of strings, Supersymmetry and Evolution of galaxies,

Introduction -:

One most important theory about the origin of our observable Universe is the cosmic strings or superheavy strings theory which are predicted to form in the early Universe in Grand Unified Theory ( G.U.Ts),in the standard Big Bang model, where loops of cosmic strings were considered the seeds of all galaxies of this Universe . Before we authors directly jump into Superheavy string theory,let us first talk a little about elementary particles in Quantum Chromodynamics ( Q.C.D) and in Quantum Electrodynamics (QED).

About elementary Particles

What is hadrons?-:

Molecules are composed of atoms. Atoms in turns are composed of electrons and a nucleus. The nucleus in its turn is composed of protons and neutrons and the hosts of very short lived particles with which they interact , are composed of yet more fundamental particles called "Quarks". Quarks are not however found in a free state in nature. They are thought to be apparently confined in the interior of the particles they are composed of. It is relatively easy to separate an atom from a molecule, an electron from a nucleus or a proton from an atom . But it is very difficult to remove a quark from inside a Proton or a Neutron

Today the elementary particles are two types

So far these two particles had shown no internal structure further

Hadrons-: It is the most complete structure. Hadrons are defined by the fact that they interact with one another strongly. The best example of hadrons are protons, neutrons, the Mesons, the pi mesons or Pions, the hyperons which do feel strong nuclear forces. In QCD, we believe that hadrons are built of smaller building blocks like Quarks, bound together by QCD forces gluons or Glue balls .

There are hundreds of known hadrons which exist widely with diverse properties and are very short lived . The observed properties spectrum of hadrons are broadly divided in two classes a) Mesons-: which have larger spin (0, 11,2) such as pions( π) ,kanon( k), and rhO( P)

b) Baryons -: which have half integer spin ( ½, 3/2) such as Proton (P) , Neutron(N) ,Lambda( Λ) ,Omega(Ω). Mesons and Baryons can be subdivided or grouped into multiplets in which the numbers are quite smaller. This classification scheme is called'' Eightfold ways". Merry Gelman

( NL) and Zewang( NL) told that all hadrons could be built-in in bound state of few fundamentals one with spin ½ constituents, called Quarks ( fermions ) .According to him mesons are bound state of a quarks (q+) and it's antiparticle (q-) while Baryons are constructed of three quarks and anti Baryons are made of three anti quarks. Thus the antiparticles have the same mass as particles but exactly has opposite quantum numbers. The Meson and Barton so constructed should have electric charge through their quarks numbers

Quark flavours-:

Up Quarks -: symbol (u);Mass ratio (1); in unit of proton charge(+⅔); Baryon(⅓); number of spin(½)

Down Quarks-: symbol (d) ; Mass ratio (2.5,); in units of proton charge (-⅓) ; Baryon (⅓) ; number of spin (½)

Strange Quarks-: symbols (s) ; Mass ratio ( 50); In units proton charge (-⅓); Baryon (⅓) ; number of spin (½)

Charm Quarks -: symbol ( c ) ; Mass ratio (1673) ; In ratio of proton charge (+⅔) ; Baryon (⅓) ; number of spin( ½)

Bottom Quarks -: symbol ( b) Mass ratio (1125); In units of proton charge (-13e) ; Baryon (⅓) ; spin(½); it has color ; yes it decays finally into charm Quarks or into Up quarks

Top Quarks -:symbol ( t); Mass ratio (3375 or 173 Gev*2) ; In unit of Proton charge (⅔) ; Baryon (⅓) ; spin (12)

So it becomes obvious that quarks, neither anti quarks particles, were ultimately massless particles ( it was however first thought quarks were massless) when they were at rest conditions or in inertia state or in condensation phase , at a point of inflation of the Universe or before the nucleosynthesis started in the universe after Big Bang . They had rather huge mass in Gev temperature and of quarks particles families, top quarks (t) have the highest mass among fermion particles. All quarks do carry functional electrical charges. The d,s,b quarks charge -⅓ in units of Proton charge while c & t quarks have charge +⅔ . Hadron may be thus labelled by their quantum numbers. Hadrons may carry other quantum numbers which was a pain taking discovery. Over the years" baryon numbers", "strangeness" and " charms' ' were named of importance. Based on these, hadrons are classified into two broad categories "Baryons" and " Mesons" which are distinguished by their decay products. All hadrons decay themselves by emitting leptons (ℓ) photons (γ), or other hadrons,the last of which in turn in the same manner. Baryons always leave a proton among their decay products.Their antiparticles called anti baryons , decay eventually into antiprotons. Mesons either decay entirely into leptons and photons or into baryons and antibaryons pairs. To quantify all these baryons are assigned as baryon numbers and anti baryons are assigned as" B-1" and Mesons are assigned as "B-0". The significance of strangeness (s) and charm(c) is regarded as measurable integer labels carried by hadrons which are labelled in strong interaction but not in weak interaction for beta decay. It is also observed that strange hadrons (s+O) were systematically heavier than non strange hadrons with otherwise identical quantum numbers. When hadrons with charm were discovered it was found still heavier

The Proton and the Neutron have nearly the same mass ( about 938 Mev) and are quite similar in many other aspects. They differ, of course, in their electromagnetic charges. All hadrons follow this pattern. They come in small families or multiplates with very similar properties but of different charges. Thus there must be a kind of symmetry in the Universe. This symmetry is called " isospin symmetry" described to the types of families and similar properties of members of a family but does not determine in which family they do exist, like proton- neutron form a doublet, π Mesons form a triplets ( charges -1,0,+1) and″ delta"(∂) hadrons form a quadruplet ( charges (-1,0,+1+2). After the discovery of strange quarks (s) in the 1950s attempts were made to incorporate strangeness into the classification system. This was because they assigned the hadrons to families with similar properties ( in particular with the spin and parity) but with different charges and strangeness. Strange hadrons weigh somewhat more than non strange siblings, so that strong interactions are not bound to strangeness, but they are near-sighted to allow us to group particles into families anyway. The concept of or notion of "Quarks " was thus introduced in 1964 by Marry Gelman ( Ref no 2 -: Gelman M , Physics letter vol8; page2141964) and Zweig ( ref no 3-: Zewig G - CERN preprint TH 401,page412, 1964) to explain why Baryons came only in multiplets of 1;8;10 and Mesons in multiplets of 9. They in their articles first proposed the classification to imagine that hadrons are composed of three types of fundamental particles called 'Quarks".They first proposed that Baryons were made of three quarks (Q 3) and Mesons are made of Quarks and antiquarks

A nutshell about Quarks flavours-:

So all hadrons are definitely composed of small numbers of more fundamental particles called quarks. Experiments with high energy electrons and neutrinos made it possible in effective ways to take the x-ray of proton structure and it revealed that protons do contain objects with exactly the properties of quarks. The quark possesses an unusual types of charge which in QCD called " color" which operates in the same ways as the electrical charge. The relativistic quantum field theory of colour induced force is called quantum chromodynamics (QCD). By analogy with mathematically similar combinations and two types of flavours of quarks which are labelled as Up (u) and Down ( d) quarks. The Up(u) and Down (d) quarks are needed to make up both Neutrons and Protons . They move ( spin) inside the hadrons with an effective mass of about 0.33Gev.To explain the properties of heavier particles further quantum numbers (like strangeness,charm and beauty particles) must be assigned to Up and Down quarks and their quarks flavours are known as "S" quarks ,"C"quarks and "B"quarks respectively ."The strange quarks" (S) with mass of 0.5Gev is needed to make up the strange particles which were discovered in cosmic rays in 1940.The "charmed quarks" ( c ) with a mass of 1.5 Gev is needed to make up z/W bosons particles."The bottom quarks" (b) with a mass of 4.8Gev and was found to make up Gamm(γ) particles. A further flavour of quark (T) or "Top quarks" is predicted to form the symmetry arrangement and CERN scientists confirmed it in 1997.The evidence for Top quarks came through a dozen or so events contains either W Bosons ( seen via their decays into an electron or muon and a neutrinos) or one such W boson together with three or four sets of hadrons. These are believed to be due to productions of Top quarks antiquarks pairs. The collider detector facility ( CDF) events have distinctive Top and anti Top quarks features. For example many of them contained bottom quark or antiquark . The CDF announced the mass of Top quarks in 171 ±16Gev. At the large electron positron collider ( LEP) the mass of many decay properties of Z bosons had been measured also enabling the mass of Top quarks 171±15Gev . Both the charm quantum numbers and strange numbers are conserved by strong and weak electromagnetic interaction. This means that charmed or strange particles must undergo a weak decay whose rate is determined by two factors: the weak interaction coupling constants and density of allowed final stage i.e. phage -space availability. The characteristic lifetime of strange particles decay are around 10*11 seconds.The charmed particles are much heavier than the strange particles and charmed particles are much heavier than u,d,or s quarks

To explain simultaneously the "Symmetries and charges " of the observed hadrons the quarks are also required to have the electrical charges,which is a fraction of the electron charges "e".Thus an UP (u) and down quarks(d) have charges+2/3e and -1/3e respectively. Thus a proton consist of 2 down ( d) and one up( u) quarks or may be designed as d2,1u

( Ref 3-: Close E. G " introduction of quarks and protons " at page P3-88; Academic press New York ,1979). The lambda particles (Λ or λ)) consist of one Up quarks,one strange quarks and one down quarks.(u1,s1,d1), similarly Omega particles (Ω) consist of 3 strange quarks or may be designed as (s3), Pion(π ) particles consist of 1 up quarks and 1down anti quarks( 1u,-1d), kaon particles (1u,s1), charmonium (J/psi meson) can be (1c,-c1), Bottomonium (n2S+1LJ) particles ( b,-b). Similarly strange hadrons are made of using s quarks. The Meson or iP particles are bound states of charm and anti charm quarks . Although the quark model was very much indeed successful in classifying the observed hadrons,in the 1970s, still it became clear that the simple quark model was inadequate. In addition to quark flavour, quark must carry additional quantum numbers which are known as " color" and each quark flavour comes in a distinct color,there are as many as color as we thought. There are red, green and blue color . So Up quarks are now denoted as ur,ug,up,ub. They are identical in all respects ( mass, charges, and so on) except in their colors. Similarly colors are present in other quarks like in down quarks ( dr; dg; db) for strange quarks (sr; sg; sb) for bottom quarks ( br;bg; bb) for Top quarks ( tr;tg:tb) and so on and so on . So color was initially three types in the earliest universe ( even when nucleosynthesis did not start) , and they were Red, Green, Blue.

The existence of strongly interacting particles with spin ½ is the quarks particles that have half integral spin and as such they ( quarks) are known as Fermions. But the basic difference from other Fermions particles from the quarks is that quarks carry extra quantum numbers " color" which the Fermions do not carry. The quarks as they carry these color and so hadrons are always colourless.

Quarks are in spin motion

In the particles and even all the structures found in the universe are in spin motion but who started spinning is the big question: the Higgs fields? -: .

Particles like electrons, neutrons, protons, and quarks are in right handed or in left handed spin movement. In QCD, we believe that hadrons are built up from smaller building blocks of quarks bound together by QCD forces. Protons contain several such quarks spinning in opposite directions. Spin effects play a very important role in the very early stage of quark structure of hadrons. Actually quark structure model gave the first satisfactory explanation of magnetic movement of the proton and neutron having their particles made from two kinds of flavours called up and down with electrical charges of +⅔ and-⅓ of the proton charges respectively.The proton is thus made of 2u quarks and 1d quarks and neutron is made of 2d quarks and 1u quarks thus giving observed charges of+1 and 0. Because all have the spin as proton,the right total spin for the proton and for the neutron is obtained by having the u and d quarks in opposite directions giving a total spin which is the difference between u-d quarks spin.

Electric charges spinning 0 in opposite directions have their magnetic movements all pointing in the same direction and added up to a large value. These explain why proton with a positive charge has a magnetic movement roughly three times the Dirac value and neutron with no charges has magnetic movement corresponding roughly to double the Dirac value for a particle with negative charges

The big question to present authors of this article is whether further sub & sub elementary particles in QED were possible than quark itself? Geonium is a man made atom created in liquid helium temperature in ultra high vacuum from an individual electron in magnetic and electric trapping fields. For this atom the electron gyromagnetic ratio g=20,000,000,000110(60) has been measured in microwave spectroscopy experiments. After substraction of quantum electrodynamics shifts , The g-gDirac=11x10*-8 excess over the value g Dirac=2 for the theoretical Dirac point electron suggest for the electron of nature a corresponding excess radius Re-R, Dirac over the Dirac radius R Dirac=0 and so it must be a special structure ( Ref 4-: Bhattacharya Rupak, Bhattacharya Ritwick, Mukherjee Dalia Bhattacharya Pranab Kumar "Sub2 quark particle possible in the origin of mass in early universe " , discussion in Extreme Astronomy.com). A near Dirac radius Re~-10*-20 centimetres must be present . In the Big Bang cosmology a near zero rest mass particles are required in singularity point which indicates from nothing state ( zero mass) resulted a spontaneous quantum jump to give finally mass ( Higgs particles are now thought to give mass to a mass less particle in Higgs field ) in earliest big bang time and these particles and antiparticles were in ½ spin ( Dirac point particles). So far quarks particles and it's sub quark energy color and Higgs particles are discovered . In 1974, the Pakistan Physicist late Prof Abdul Salam ( Nobel laureate in physics) and his associates pictured the electron, a particle on the level of a quark as composed of three subquarks each 10*10 times heavier than electron in Gev mass, as like a proton is composed of three spin ½ particles.This is now nomenclatured as "Salam particles' according to late Prof Abdul Salam (who received Nobel prize in physics as citizens of Pakistan).The big question remain to present authors is it possible to have sub2 quarks particles with zero rest mass in Gev temperature?" . This particle must be sub-sub x quarks, particles less and less imperfect near Dirac particles held together by new stronger and stronger forces and with almost zero mass in Gev. Probably to the beginning of our universe in planck epoch such a particle existed where Higgs particles fused to give mass in Higgs fields

Forces regulating our universe -:

Now there are three important forces that regulate our observable four dimensional universe ( space time is when taken as 4th dimension).They are Gravitational forces, Electromagnetic strong forces and electromagnetic weak forces. The gravitational force is well understood classically since Newton's theory was propagated by Einstein. But even today there is no satisfactory particle theory for the gravity in particle form except a " Graviton"(G) particle (mass in Gev 0 < 6×10*−32 eV/c*2) . The effect of gravity is quiet negligible in experimental particle physics and Gravity is very important for large bodies.The electromagnetic force holds atoms together and control the behavior of matter.The strong force is responsible for holding the nucleus together and weak electromagnetic force is responsible for beta decay and other rare process. Now once it was known that hadrons are composed of quarks ,it became possible to construct the theory of strong force.

It is the force that rather acts between quarks which hold together to form hadrons generating nuclear binding. In QED ,for so long the electromagnetic forces were generated by exchange of photons between electrically charged particles. Weak forces were generated by exchange of new particles H, W,and Z particles. But in QCD , the inter quark force requires another particle called "Gluons' ' in gluons fields. Both the quarks and Gluons carry color charge. So Gluon exchange generates Gluon -Gluon force as well Gluon -quark or quark-quark forces . What are the Gluon particles in QCD? Are then gloun sub -sub quark particles?.In certain circumstances quarks radiate energetic particles further,which give rise to characteristic controlled YETS of Gluon particles. Such YETS were discovered in electron proton colliders called PETRA in Hamburg which revealed the existence of Gluon particles in their Nature. Gluons are theorised really massless particles with spin 1 and negative parity ( charge 0, spin 1, mass in Gev/c*2 is 0 force mediated Strong force) . Gluons are mixtures of two colors, such as red and antigreen, which constitutes their color charge. The evidence of Gloun is more quantitative and theoretical for evidence for quarks particles.About half of the momentum of the nucleon is carried by quarks while objects which do not interact either electromagnetically or weakly,must then carry the remaining momentum. It is the Gluon. Gluons can be from quarks by strong interacting process. Perhaps most direct evidence for Gluon particles came from observation of jet like structures in high energy annihilation of e+e- experiment. Colliding beams of e+ and e- particles have been observed to annihilate into hadrons, most of which were found from two YET-like structures in PETRA experiment. The Gluons or Glue balls can be also identified in R ( Rupak particles) particles . Their measurements of the lack of electromagnetic couplings of R particles led the present authors to hypothesize that R particles (Rupak particles) consist of Gluon bundles. In QED photons transmit the electromagnetic forces . Following the discovery of Psi( An electrically neutral meson having a mass 7,213 times that of the electron and a mean lifetime of approximately 1 X 10*-20 seconds, composed, like the J/psi particle, of a charm-anticharm quark pair particles) , a theory released that implied that radiative Beta (β) decay of Psi particles should be the good sources of Gluon matter( in ordinary matter there should be existence of Glue balls also. ) Matter here is made of gluons as most finest particles, but obviously then matter is not composed of only quarks as finest one. Rather we can say R particles are prima facie candidates for resonating Gluons

Cosmic Strings

With this very superficial little background knowledge of particle physics, let us now discuss the Super heavy string theory of galaxies evolution ,The main alternative theory of origin of structures of the universe are cosmic strings or super heavy strings which are predicted to form in the early Universe by Grand Unified Theory ( GUT) in inflationary Big Bang model . Loops of cosmic strings were actually the seeds of galaxies. They were super heavy strings,formed at phase transition or condensation that took place when the universe cooled after GUTS in the very early universe . Kibble ( NL in physics) had suggested that GUTS strings played an important role in the evolution of the universe and the strings provided the inhomogeneity leading to formation of galaxies. In the very early universe strings were predicted to be formed at symmetry breaking phase transition by those Grand Unified Theories,in which vacuum had the appropriate topology . Cosmic strings are the configuration of the matter fields which owe their topology of the space of degenerate vacuum , produced by the phase transition in the early Universe . Let us ignore the normal structure of the strings and treat them as one dimensional objects with tension. In the resting frame of strings,the mass per unit length ∞ to tension . The equality of the line of density and the tension caused the typical velocity associated with large vibration on the strings to be close to the speed of light . The strings can not end but can either close on themselves or can be extended to infinity (∞). The closed strings formed the string loops. As with any objects in tension, strings would also accelerate, so as to try to become straight .Damping of the strings motion was due to their gravitational interaction with other matters , those become negligible as soon as the strings were formed. Strings that extended outside the horizon of the universe were conformably stretched by the cosmic expansion. Thus at a given epoch these strings were straight on their length and scale, but were smaller then the horizon size, but was quite convoluted on a large scale then this. The typical velocity was associated with straightening of a string and was close to speed of light and the velocity field of the string extending outside the horizon was relativistic and approximately constant over scale much smaller than a horizon size. Once a loop entered the horizon it no longer expanded but rather started to oscillate with a period comparable to light travel time across it. The motion was damped by gravitational radiation causing the size and period of loops to decrease approximately linearly with time. The fractional decrease in size , period,and mass of the strings in one oscillation was given by equation Guw where G is Gravitational constant. A string will decrease to zero size in a finite amount of time losing its energy by only gravitational radiation. The distribution of strings in our universe were not quite so often well understood. After the phase transition the strings were formed in a random network of self avoiding curves/ loops.Some of the strings were in closed loops and some were as infinite strings.The distribution of strings so happened that a constant number of loops entered the horizon. If the infinite strings would simply straighten out , then numbers of open strings across the horizon-sized volume would also increase with passing time and strings would soon come out to dominate the density of the universe . But it didn't happen. Velenkin ( Ref no 5-: velenkin A, Physics Review D23; P852: 1981) showed that the geometry produced by the gravitational field near a length of straight string is that of Minkowski space with three dimensional ( 3D) l wedge taken out of each space, like sliced . The vertex of the wedge lies along the length of the strings and angle subtended by missing wedge lies in the frame of the strings and is equated as ∂πGu. The two exposed faces of the strings thus could be identified. Thus space time remained everywhere except along the strings,where it was highly curved .If Gu<<2,then the stress energy of the strings would produce only small ( lenier) perturbation from the metric of the rest of the universe. Because the matter hadrons in the universe didn't produce significant perturbation from the Minkowski metric space,on scale,less than horizon, the gravitational field at a point ,much closer to a length of a string would be essentially then the same as gravitational field at similarly located point in Minkowski Space. In the rest frame of the strings ,all elementary fundamental particles were, when passing,the strings deflated by an angle 8πGu with respect to all particles passing on the other side of the strings. The magnitude of discontinuity in temperature across the strings ( while passing of particles) was ∂T/T=8πGb where G =Gravitational constant ,b=transverse velocity of strings which typically was close to unity. This jump of temperature persisted on angular distance away from the strings, corresponding to the present angular size of the radius of curvature of the strings.The magnitude of temperature jump was then independent of Red shift (Z) at which light rays reaching to us, passed by the strings. If we calculate the general properties of microwave sky anisotropy in string mode then let us assume that microwave photons were last scattered at Redshift Z=1s. So in a perfectly homogeneous universe the matter became mostly neutral and optically detectable than at Redshift Z ~1000. However in an universe,with strings, there will be large amplitudes in homogeneity on a small scale and the heat output from objects forming at or before Z/τee may re-ionize the plasma. If the plasma were fully ionized then Z1s>10 and we have 1000>Z1s>10,the angle subtended by a horizon sized volume space at Z1c is θ1s*-½<<1. One would expect to see on a round patch of sky of strings per horizon volume at Red shift Z will project one length of string of angular size θ if z<1s. These strings will then oscillate moving relativistically as they were unable to straighten themselves out of this length scale.In modern Gauge theories of fundamental interaction of vacuum was far from nothing. Rather it is now recognised as a dynamical object that was in a different state. The state of vacuum affected the properties such as masses and interaction of particles put into it.

vacuum was created -:

Although the vacuum is thought to be in a ground state,with the lowest state of energy, this state has not always been the same. Thus in early universe when the particles components ( ordinarily matter and radiation) was at very high temperature,the vacuum adjusted it's state in doing so , modified the properties of particles so as to minimise the free energy of entire system ( vacuum plus particles) i.e the vacuum went into higher energy state in order to lower the energy of hot plasma by even greater ammount.As the universe cooled to keep the entire system at the lowest possible energy,at a given temperature,the vacuum had to change eventually ,ending up to its present state which is nearby the true or zero temperature vacuum. It was possible in early universe that as the universe expanded, the cooling happened too rapidly for the vacuum to find its true ground state and the vacuum was frozen into ground state with defects.Defects that probably could occur in a three dimensional space could be zero dimensional ( Monopoles),Two dimensional ( domain walls) or one dimensional ( Strings). The strings are so microscopic objects.In most cases of cosmological interest they have no ends and are either infinitely long or they are in closed loops . At GUT, the strong,weak and electromagnetic forces behaved as if they had equal strength much as line defects found in crystals.They formed as a network across the space time.The GUTs predict that strings were formed at a temperature of about 19*15 to~ 10*16 Gev, at cosmological time of 10*~35 second. The cosmic strings were formed at the mass scale of GUTs symmetry breaking ( Mx-2x10*15Gev) was typified by mass per unit length uG/c*2~2x10*6 in …… ( G = Gravitational constant,c= speed of light which is corresponding to u =

The strings were formed with a mass per unit length of about 10*20 kg*-1. Because of their enormous tension e/G the net work of the strings were formed in phase transition.In this theory the strings contributed only a small fraction of mass of the universe.The Galaxies were formed by accreacting of ordinary matter around the strings.The strings were attached by subsequent expansion of universe on wave on a given scale and began to oscillate then .The strings underwent oscillation in which transverse inertia acted as weight and the restoring factors were provided by longitudinal tension of the Strings. The gravitational fields of these string loops caused accretion of matter around them . Brosche PG in the journal of Astrophysics, stated that angular momentum of an astronomical object is proportional directly to square of mass and constant of proportionality is comparable to string theories,which suggests that the universe had evolved through hierarchical breaking of rotating or oscillating strings and the angular momentum with mass between various classes of different objects ranging from planets to superclusters

( Ref no 6 Brosche PZ " j.astrophysics vol 57; P 143; 1963) . For the past four decades a variety of Grand Unified Theory ( GUTs) had been developed to unify the strong and weak interactions at an energy scale of 10*16Gev .GUTs are gauge invariant point fields theory of Yang and Mills, the two Nobel laureates in physics which do not incorporate gravitational forces and henceforth there remains few theoretical constraints on the possible internal symmetry group as per these authors of this article.

Super symmetry? -:

The most favourite GUT theories are based on the special unitary group Su(5), the special orthogonal Group SO(10) or the exceptional Group E6. In such GUT theories quarks particles and leptons particles make up three of these families that are unified in one' family. Super symmetries are an important ingredient of GUTs. It is symmetry that relates to Fermions and particles of different spins. But Supper symmetry is not an internal symmetry of the universe, but amounts to an extension of space-time in super- space that includes extra spinorial anti-commuting co-ordinates as well as ordinary coordinates. Super symmetry requires particles such as s quarks , s leptons, winos,zinos,R particles which have yet to be discovered. Super Gravity theories are point field theories that incorporates local or gauged supper symmetry and thereby enlarging Einstein theory of relativity. The basic idea of gauge theory is a continuous symmetry or global invariance properties of Lagaragian field theory that can be made into a local invariance by gauge field theory. This means that given a field theory, which proposes symmetry such as U1(1) ,Su(2),Su(3) or any other U group. The theory can be extended in a gauge theory which has the symmetry at each part in the space time individually. The new symmetry is then called Gauge symmetry because it implies we can choose our measuring standard Gauge to differentiate throughout space time without changing the physics of the theory. The most familiar example of a Gauge theory is electromagnetism . In QED , the quantum field theory of electromagnetic interaction are charged particles and photon / Boson is the most successful gauge theory.The behavior of a relativistic string moving in space time differs significantly from that of a point particle. Unlike a point particle , a classical relativistic string has an infinite number of vibrational modes with arbitrarily high frequencies and angular momentum. This means that in quantum theory , a string has an infinite number of states with masses and spins which increases without limit.

String theory developed in the 1970s as model of strong interaction physics. A meson has thought of as a string with a quark attached to one end while an antiquark to the other end. The string tension (T) was supposed to be ~1Gev*2 and the excited states of the strings were supposed to be hadrons. The main theories were " Boson theory" which only described Bosons and the spinning theories that incorporated Bosons as well Fermions . These early string theory had several theoretical inconsistencies according to present authors of this article, because the strings ground states always turned out to be Tachyons ( Reference no 6-: Rupak Bhattacharya, Pranab Kumar Bhattacharya, Upasana Bhattacharya, Ritwik Bhattacharya Rupsa Bhattacharya, Dalia Mukherjee etal 'Tachyon faster than light particles exist in our universe or an imaginary mathematical particles " International journals of Astronomy Astrophysics and Space Science vol 2; no 3; June 2015; page 12-29). Superstring theory that evolved from spinning string theory that incorporated Supersymmetry and had no tachyonic ground state. Superstring theories hence offered the possibility of constructing a consistent quantum theory that unifies all interactions including gravity and natural mass scale set by string tension (T) at Planck's scale ( T*½=10*9Gev).The excited states were so massive that they could be taken to be infinitely heavy ( like superclusters of galaxies) and the theory can be approximated by effective point field theory of the mass-less state only. At the energy scale below the Planck scale, the string looks like a point. One of the constraints in any string theory is that all string theory contains a massless spin 1 and spin 2 particles which are associated with Yang and Mills Gauge Boson and Gravitation. Furthermore the original Bosonic string theory required 26 space time dimensions where as superstrings theory demands only ten (10 ) dimensional space time. We live only in a three (03) dimensional universe and we can best imagine four (04) dimensional space time. Then where are these extra six (06) dimensions in superstring theory or extra twenty two (22) dimensions in Bosonic string theory? Maybe these dimensions are curled up or coiled up and finally became very small by compactification in superstring theory or may be hidden in black hole.

Types of superstring theories -:

There are three types of superstring theories . Type 1 super string theory describes the dynamic of open strings that have their free end points. The strings here carry quantum numbers in the n dimensional, defining representation of a classical group G-S0(a) or the simplistic group USP(a) at their endpoints SU(a). This is similar to the way in which quarks quantum numbers were incorporated in the original string picture of Mesons.

The string is locally invariant under two Super Symmetries n=2 and the free ends boundaries conditions break down this just to one( N=1) supersymmetry. The mass-less open strings state is the usual state of SuperSymmetry of Yang Mills theory in ten (10) dimensions with Gauge group G . The two open strings can interact when two ends touch and join to form one open string or co inversely one string can split in two. One important thing is that the two end points of a simple string can join to form a closed string and thus massless state of a closed string forms a super Gravity and don't carry yang Mills quantum numbers

Type II superstring theory only involves the closed strings . This can have an orientation associated with the fact that waves can run around the strings in two possible directions. The two orientations allow for two chiral supersymmetries. So these theories are invariant under ten( 10)dimensions. In type II (a ) theory the two superchargers have opposite chirality and so the theory has actually no chirality and therefore it is not a very important theory. It is rather a low energy point field theory whose dimensions are D=10.n=2 non chiral super Gravity. Type II(b) theory has super charges of the some chirality. It is also low energy point field theory, the limit is the chiral dimension D=10 n=2 super Gravity. This theory is a remarkable theory in being Chiral and yet not having any gravitational anomalies and IIa very important thine of any superstring theory is that it must give rise in observed chirality of our three dimensional world which means it must maintain parity with laws of physics

Type III super string theory ,also called Heteriotic string theory ór Bosonic string theory .It is also based on closed strings only, although it carries a Yang Mills Gauge Group G and has super symmetry n =1. In this theory, instead of Yang Mills charges residue at the ends of the string there is a charge density along the whole string (Ref no 7-: Gross David , Hurvey j Martinec F, Rohm R et al Physics Review Letters vol 54; P 502; 1984 David Gross awarded Nobel prize in physics for this published work ) . This type of superstring by David Gross et al based on either E8xE8 of type I superstring theory or S0(32). It has been speculated in this theory, if the Gauge group G is E8xE8 -the E8 symmetry may persist even in dimensionally reduced theory and in that case there will be creation of two types of matter whose interaction will be described as E8. One type of matter will exactly mirror the other i.e. matter and antimatter or shadow matters will be created in this theory . This shadow matter or antimatter interacts gravitationally with ordinary matter creating huge bursts of energy and radiations . But the Heteriotic string theory involves originally twenty six (26) dimensions with sixteen (16) dimensions being maximum tours of one or other of two groups and thus this theory leaves us últimately in ten (10) dimensional space time.

It was again Kulza in 1922 and Oscar klem in 1926 who jointly showed that if a person assumed general relativity in five dimensions, where one dimension was curled up , the resulting theory then would have a four dimensional theory of electromagnetism and Gravity. Electromagnetism energy there remains as Gravity in the curled up 5th dimension. Wien had identified the momentum of particles moving around the 5th dimension as gravitational charge . Bosonic string theory ( type III) requires 26 space time dimensions when super string theory contains 10 space time dimensions. So in string theory at least there are six (6) or seven (7) extra dimensional space time remaining. One can imagine that these extra dimensions are curled up to form a small manifold and remarkable such six or seven dimensions compactification can produce a world remarkably like our own World in which the shape of extra dimensions determine the matter content and forces of nature as seen in four dimensional observers.

So the strings could occur as vortex in Gauge theories in anomalies manner in the formation of Abrikosov megantic filaments in super conditions and such a vortex was formed in phase transition in the very early universe in the Higgs fields. . The strings then broke down or chopped of mesh at speed of 10*8 seconds after the Big Bang.This spontaneous broken theory with Higgs fields says 1M* 2X/4£

G∫πn*2∫G where π is true vacuum expectations value of Higgs field and MX Mass of the associated vector boson G=g*2/4π is the Gauge field coupling constants ( G is Gravitational fraction for the Abelian Higgs Model .If MH=MX where MH is the mass of scalar Higgs particles' then ∫G=1 then electro week with u=19*2and MX=100Gev and the string mass Ucw~-2x10*-5 g cm*1 ~=2x10*-33 c*2/G . For Grand unification strings alpha G ~10*-2and MX ~10*-15Gev yelding uGv gamma ~2X10*21GCM*-1~=2X10*-7c*2/G where C is Speed of light ( Ref no 8 Velkin A Physics Review Letter Vol 46; Page 1169; 1981) which shows that value of u for astronomical strings much closer to GUTs strings.

Evolution of Galaxies stars planets from cosmic strings -:

So Brosche and Tassie ( reference no 9 LT Tassie Nature vol 323 P 40 1986) theory suggested a very different evolution of the universe and Galaxies. According to them , some times in the early past of the universe, strings constituted all or nearly all of mass of Universe and that astronomical objects we today see originally formed from the strings in such a way that a large piece of string which eventually corresponded to Superclusters galaxies ( it contains some millions of galaxies ) broke into smaller pieces of strings corresponding to clusters of galaxies . These pieces of strings in turn further broke into further smaller pieces of strings corresponding to galaxies . They even broke into further smaller pieces of strings to stars and so on and so on. At each stage of hierarchy breaking of strings the new pieces might have some vibrational energy and the vibrational energy was large compared with mass to come on. Eventually the pieces of strings transformed either by phase transition or some form of rapid breaking into ordinary matter and thus became planet satellites, asteroids etc that we see now . But many other theory say that seeding of galactic matter and radiation densities, the universe passed the state of equal matter and antimatter and radiation densities some about 10*11seconds (30000 years) later than galactic string loops chopped off mashes after 10*8 seconds after the initial Big Bang .

So the idea that super heavy strings were formed at phase transition in the very early universe provides an explanation of the origin of galaxies. GUTs formed strings at symmetry breaking phase transitions in the early Universe in which vacuum ( voids) was appropriate topology. They have well known analogues in condensed matter physics. Much as the line defects the strings formed a network across the whole space . In some theories strings were formed after a period of inflation. They were then stretched by subsequent expansion of the universe and waves on a given scale began to oscillate as the scale entered the particle's horizon. Wherever the strings crossed itself,an exchange of partners occurred and produced crossed oscillating loops of strings with long life times. The gravitational force of these loops caused matter to accrue around strings. Thus the strings could be the primordial density fluctuation needed in the early Universe to explain eventual formation of galaxies. Now the question stands how such strings would produce density fluctuation on a broad range of scale which was responsible for formation of galaxies? Galaxy formation from the dark matter was an extremely active area of study. From the Big Bang nucleosynthesis ,it was known that Baryons accounted for less than 15% of critical cosmological density. Observation of the dynamics of galaxies suggests that matter clustered with galaxies is 60% of the critical density ( K) and the lower end of observed range is consistent with the Baryon limit. Thus the dynamical Dark Matter could be Baryon?There are however two arguments which drive us to look for other candidates.The first is inflation and the second is anisotropy of 3K microwave background.

Inflation was the only way of explaining several otherwise extraordinary initial conditions of the universe.But for fine tuning of inflation required a critical density of the universe . Thus at least 85% of the universe (15% were Baryon) could not be the Baryons matter and more than 60% of the matter of this universe so didn't cluster into galaxies.The density of the matter on the universe must be greater than baryonic upper limit . To make things a little more difficult,it is said that " the special correlation functions of rich clusters of galaxies had revealed strong clustering of very large scale up to 150 MPC" .This correlation functions of clustering of galaxies was 18 times stronger than the special correlation functions of galaxies. It was also found that the largest scale of the universe seems to look filamentous ( strings are here turned filaments ) with large vacuum ( voids) and large clumps .With the GUTs an excellent way appeared to produce the distribution of size of universe. In normal generation and application of GUTs ( a fluctuational spectrum with equal powering all scales formed naturally) it was assumed that there was no special correlation between large scale and small clumps.They each had random probability of occurring anywhere in the universe. On the other hand it means that strings are still produced in some spectrum,somewhere and in some size. Different proposals so had been put to solve the problem but no models could solve it as long as it was assumed that primordial fluctuation had random phases. For example a model based on Neutrinos produced both critical density and large scale structures like filaments voids clusters correlation functions but didn't account for early formation of galaxies ( Ref no 10 Bachcall N j astrophysics vol270; P20;1983).Models evoking heavy or slow moving particles[ like Gev mass photinos, gravitons, axions, planetary mass black holes] however fits the small scale structure galaxy correlation functions, formation of time and so forth as well as building hierarchical to yield clusters , but it do not allow critical density of the universe to be reached. Even the hybrid model with low mass and huge mass ions also runs into problems, because low mass particles smear out of the small structure of the universe.A more natural solution of the problem might be non random phases of the string model. JE Peebles Nobel laureate in physics ( Ref no 11 JE Peebles Nature vol 311.P517;1984) noted that non random phases of string model of the universe yields large scale filaments and voids as superheavy strings attract galaxies and clusters and give string- clusters -strings correlation.Work by JE Peebles showed that a model based on clustering of galaxies about filaments ( here strings) fit higher 3and 4 points correlation functions for galaxies as well as hierarchical clustering. This model also enables density growth in some areas without producing a large universal background anisotropy and so could enable Baryons to be Dark Matter on galaxy and cluster state with non baryonic stuff being a critical density background. The degree of Random to non random phases in such a model depends upon the density of strings in the space . In the limit of space being completely filled with strings the string picture also give random phases. Even if string densities are large enough to randomised phases their mere existence would still alter galaxy in formation , calculation, because it were the strings rather than matter that would carry the fluctuations.

The universe's rhymes ( References 12) —

The appearance of all similar structures in different areas of physics—underlie the ways theory can potentially unifies gravity with the other forces of nature and thus can eliminate the ultraviolet divergences that plague quantum gravity. String theory has, even among theoretical physicists, the reputation of being mathematically intimidating. But many of its essential elements can actually be described simply. Our this article aims to answer a few basic questions about the string theory and evolution of our most distant galaxies in the earliest time of the observable universe. 1) How does string theory generalize standard quantum field theory? 2) Why is string theory forcing us to unify general relativity with the other forces of the universe, while standard quantum field theory makes it so difficult to incorporate general relativity? 3) Why are there no ultraviolet divergences in string theory? 4) What happens then to Einstein’s conception of spacetime in string theory?

Any physicist who studied at a very high level of physics is aware that although physics—does not precisely repeat itself, it does rhyme, with similar structures appearing in different areas. We can say that Einstein's gravitational waves are therefore analogous to electromagnetic waves or to the water waves at the surface of a pond when a stone is thrown in water . We authors here will begin with one of the universe's rhymes: an analogy between quantum gravity and the theory of a single particle. Even though we do not actually understand it well, quantum gravity is supposed to be some kind of theory in which, from a macroscopic point of view, we average, in a quantum mechanical sense, over all possible spacetime geometries. (We do not really know to what extent that description is valid microscopically in string theory.). So Gravity plays a more important role always in macroscopic objects rather than it plays on microscopic or in a single particle. The averaging if performed, in the simplest case, with a weight factor exp (iI/ℏ), where I is the Einstein–Hilbert action:

I=1/16πG ∫d*4x√g (R−2Λ).(*= As super script ie to the power of )...... (equation no1) where G is Newton’s constant, g is determinant of the metric tensor, R is curvature scalar, Λ is a cosmological constant, and d*4x is the spacetime volume element.

Let us try now to make such a theory with one spacetime dimension instead of classical four space time dimensions. The choices for a one-manifold are quite limited:

Picture (1)

⁹

Moreover, the curvature of the scalar field is then identically zero in one dimension, and all that’s left of the Einstein–Hilbert action is the cosmological constant. However, Einstein’s fundamental insights were not tied to the specific Einstein–Hilbert action. Rather, they were in the broader sense that spacetime can vary dynamically and that the laws of the universe are generally covariant, or invariant under arbitrary diffeomorphisms (coordinate transformations) of spacetime.

By applying those insights, we can make a nontrivial quantum gravity theory in one dimension provided we include matter fields.

Adding matter in scalar field -:

The simplest of matter fields are scalar fields X**I, where I = 1, … , D. (** Is underscript )

The standard general relativistic action for scalar fields then is

I=∫dt√g [1/2D∑I=1 g*tt (dX**l/dt)*2*−1/2m*2],......(equation no 2)

where g**tt is a 1 × 1 metric tensor, and the Λ term has been replaced now with m*2/2.

Let us now introduce the canonical momentum P**I = dX**I/dt. The Einstein field equation—which is the equation of motion obtained by varying the action I with respect to g— is just g*tt D∑I=1P2/I+m* 2=0. ……(equation no 3)

We if pick the gauge g*tt = 1, so the equation is P*2 + m*2 = 0,(equation no 4) with P*2 = ∑IP I*2. Quantum mechanically (in units with h= 1), PI = −i∂/∂**XI, and the meaning of the equation P*2 + m*2 = 0 is that the wave function Ψ(X), where X is the set of all X**I, must be annihilated by the differential operator that corresponds to P*2 + m*2. Then

(− D∑I=1 ∂*2/∂X*2** I

+m*2)Ψ(X)=0. ….(.equation no5)

This is an equation—the relativistic Klein–Gordon equation in D dimensions—but in Euclidean signature, in which time and space are on equal footing. To get a sensible physical interpretation, we should now try to reverse the kinetic energy of one of the scalar fields XI so that the action becomes

I=∫dt√g{½g*tt [−(dX0/dt)*2+D−1∑i=1(dXi/dt)*2]−1/2m*2}....(.equation no 6)

Now we know that the wave function obeys a Klein–Gordon equation in Lorentz signature: and thus

(∂*2/∂X*2 0−D−1∑i=1∂*2/∂X*2i+m*2)Ψ(X)=0……(equation no 7)

So we have found an exactly soluble theory of quantum gravity in one dimension space time that describes a spin-0 particle of zero (0 ) rest mass m propagating in D-dimensional Minkowski spacetime. Actually, we can replace Minkowski spacetime by any D-dimensional spacetime M with a Lorentz (or Euclidean) signature metric GIJ, the action being then

I=∫dt√g(1/2gttGIJ dX*I/dt dXl/dt−1/2 m*2).......(equation no 8)

Figure 1a

Figure 1a caption

.[ A graph with trivalent vertices.

The natural path integral to consider is one in which the positions x1, … , x4 of the four external particles are fixed, and the integration is over everything else. A convenient first step is to evaluate an integral in which the positions y1, … , y4 of the vertices are also fixed. This Feynman diagram can generate an ultraviolet divergence in the limit that the proper-time parameters τ1, … , τ4 in the loop all vanish.]

From here on, summation over repeated indices is implied. The equation obeyed by the wave function is now the massive Klein–Gordon equation in any curved spacetime M:(−G*IJ D/DX*I D/DX*J+m*2)Ψ(X)=0,...( equation no 9)

where D represents covariant differentiation.

Just to make things more familiar, let us authors further go back to the case of flat spacetime . Let we calculate the probability amplitude for a particle to start at one point x in spacetime and end at another point y. We can do it so by evaluating a Feynman path integral in our quantum gravity model. The path integral can be done over all metrics g(t) and scalar fields XI(t) on the one-manifold with the condition that X(t) is equal to x at one end and to y at the other.

The process of evaluating the path integral in our quantum gravity model is to integrate over the metric on the one-manifold, modulo diffeomorphisms. But up to diffeomorphism, the one-manifold has only one invariant, its total length τ, which we will interpret as the elapsed proper time. In our gauge g*tt = 1, a one-manifold of length τ is described by a parameter t that covers the range 0 ≤ t ≤ τ. Now on the one-manifold, we have to integrate over all paths X(t) that start at x at t = 0 and end at y at t = τ. That is the basic Feynman integral of quantum mechanics with the Hamiltonian being H = ½(P*2 + m*2). According to Feynman, the result is the matrix element of exp(−τH) is

G(x,y;τ)=∫d*Dp/(2π)*D exp [ip⋅(y−x)]exp[−τ/2 (p*2+m*2)]..... equation no( 10)

But we have to also remember to do the gravitational part of the path integral, which in the present context means to integrate over τ.The integral over τ gives our final answer:

G(x,y)=∫*∞0dτ(G(x,y;τ)=∫d*Dp/(2π)*D exp[ip⋅(y -x)]2/p*2+m*2…(.equation no

11) This formula is the output of the complete path integral—an integral over metrics g(t) and paths X(t) with the given endpoints, modulo diffeomorphisms —in our quantum gravity model. The function G(x, y) is the standard Feynman propagator in Euclidean signature, apart from a convention-dependent normalization factor. Moreover, an analogous derivation in Lorentz signature (for both the spacetime M and the particle world line) gives the correct Lorentz-signature Feynman propagator.

So we have now already interpreted a free particle in D-dimensional spacetime in terms of 1D quantum gravity. can we include interactions? There is actually a perfectly natural way. There are not a lot of smooth one-manifolds, but there is a large supply of singular one-manifolds in the form of graphs, such as the one in figure 1. Our quantum-gravity action makes sense on such a graph. We simply take the same action that we used before, summed over all the line segments that make up the graph.

Now to do the quantum-gravity path integral, we have to integrate over all metrics on the graph, up to diffeomorphism. The only invariants are the total lengths or proper times of each of the segments. Some of the lines in figure 1 have been labeled by length or proper-time variables τi.

The natural amplitude to compute is one in which we hold fixed the positions x1, … , x4 of the graph’s four external particles and integrate over all the τi and over the paths the particles follow on the line segments. To evaluate such an integral, it is convenient to first perform a computation in which we hold fixed the positions y1, … , y4 of the vertices in the graph. That means all endpoints of all segments are labeled. The computation that we have to perform on each segment is the same as before and gives the Feynman propagator. The final integration over y1, … , y4 imposes momentum conservation at each vertex. Thus we arrive at Feynman’s recipe for computing the amplitude associated with a Feynman graph—a Feynman propagator for each line and an integration over all momenta subject to momentum conservation.

A more perfect rhyme-:

We have arrived at one of nature’s rhymes. If we imitate in one dimension what we can expect it to do in four dimensions to describe quantum gravity, we shall end up with something that is ordinary quantum field theory in a possibly curved spacetime. In our example in figure 1, the ordinary quantum field theory is scalar ϕ3 theory because of the particular matter system we started with and because our graph had cubic vertices. Quartic vertices, will give ϕ4 theory, and a different matter system will give fields of different spins. Many or maybe all quantum field theories in D dimensions can be derived in that sense from quantum gravity in one dimension.

There is actually a much more perfect rhyme if we repeat the procedure in two dimensions—that is, for a string instead of a particle. We immediately run into the fact that a two-manifold Σ can be curved:

Figure

On a related note, 2D metrics are not all locally equivalent under diffeomorphisms. A 2D metric in general is a 2 × 2 symmetric matrix constructed from three functions:

g**ab=( g11/g21 g12/g22),g**21=g**12…

..equation no( 12)

A transformation of the 2D coordinates σ, generated by σ*a→σ*a+h(σ),a=1,2……(equation no 13) ,can remove only two functions, leaving the curvature scalar as an invariant. All those suggest that the integral over 2D metrics will not resemble what we found in the 1D case. But now we notice the following. The natural analog of the action that we used in one dimension is the general relativistic action for scalar fields in two dimensions, namely

I=∫d*2σ√gg*abG**IJ∂X*I/∂σ**a/∂X"J/ ∂σ**b.

…..(equation no 14) . But this is conformally invariant, that is, it is invariant under a Weyl transformation of the metric gab → eϕgab for any real function ϕ on Σ. This is true only in two dimensions (and only if there is no cosmological constant, so we now omit that term in going to two dimensions). Requiring Weyl invariance as well as diffeomorphism invariance is enough to make any metric gab on Σ locally trivial (locally equivalent to δab), similar to what we said for one-manifolds.

Figure

Some very pretty 19th-century mathematics now comes into play. A two-manifold whose metric is given up to a Weyl transformation is called a Riemann surface. As in the 1D case, a Riemann surface can be characterized up to diffeomorphism by finitely many parameters. There are two big differences: The parameters are now complex rather than real, and their range is restricted in a way that leaves no room for an ultraviolet divergence. We will return to that last point later.

But first, let us take a look at the relation between the 1D parameters and the 2D ones. A metric on the Feynman graph in figure 2a depends, up to diffeomorphism, on three real lengths or proper-time parameters τ1, τ2, and τ3. If the graph is “thickened” into a two-manifold, as suggested by the figure, then a metric on that two-manifold depends, up to diffeomorphism and Weyl transformation, on three complex parameters τ̂1, τ̂2, and τ̂3.

Figure 2b gives another illustration of the relation between a Feynman graph and a corresponding Riemann surface.

Figure 2. Caption

[From lines to tubes. (a) A Feynman diagram with proper-time parameters τ1, τ2, and τ3 (top) can be turned into a corresponding Riemann surface (bottom) by slightly thickening all the lines in the diagram into tubes that join together smoothly. The Riemann surface is parameterized, up to coordinate and Weyl transformation, by complex variables τ̂1, τ̂2, and τ̂3. (b) The same procedure can turn the one-loop Feynman diagram (top) into its string theory analog .]

We used here 1D quantum gravity to describe quantum field theory in a possibly curved spacetime but not to describe quantum gravity in spacetime. The reason that we did not get quantum gravity in spacetime is that there is no correspondence between operators and states in quantum mechanics. We considered the 1D quantum mechanics with action

I=∫dt√g(1/2g*ttG**IJ dX*I/dt dX*J/dt−1/2m*2).....(equation no 15)

What turned out to be the external states in a Feynman diagram were just the states in that quantum mechanics. But a deformation of the spacetime metric is represented not by a state but by an operator. When we make a change δG**IJ in the spacetime metric G**IJ, the action changes by I→I+∫dt√g𝒪, …..( equation no 16) where 𝒪 = ½g*ttδG**IJ ∂tX*I∂**tX*J ……(equation no 17) is the operator that encodes a change in the spacetime metric. Technically, to compute the effect of the perturbation, we include in the path integral a factor δI=∫dt√g𝒪, integrating over the position at which the operator 𝒪 is inserted.

A state would appear at the end of an external line in the Feynman graph. But an operator 𝒪 such as the one describing a perturbation in the spacetime metric appears at an interior point in the graph, as shown in figure 3a. Since states enter at ends of external lines and operators are inserted at internal points, there is in general no simple relation between operators and states.

figure 3

[Figure 3. States and operators. (a) A deformation of the spacetime metric corresponds to an operator 𝒪 that can be inserted at some internal point p on a Feynman graph. By contrast, a state in the quantum mechanics would be attached to the end of one of the outgoing lines of the graph. (b) A Riemann surface can also have an operator insertion. (c) If the marked point in panel b is deleted, the Riemann surface is conformally equivalent to one with an outgoing tube that is analogous to an external line of a Feynman graph. The operator 𝒪 that was inserted at p is converted to a quantum state of the string that propagates on the tube.]

But in conformal field theory, there is a correspondence between states and operators. The operator 𝒪 = ½g*abδG**IJ∂**aX*I∂**bX**J ..…..(equation no 18 )that represents a fluctuation in the spacetime metric automatically represents a state in the quantum mechanics. That is why the theory describes quantum gravity in spacetime.

The operator–state correspondence arises from a 19th-century relation between two pictures that are conformally equivalent. Figure 3b shows a two-manifold Σ with a marked point p at which an operator 𝒪 is inserted. In figure 3c, the point p has been removed from Σ, and a Weyl transformation of the metric of Σ has converted what used to be a small neighborhood of the point p to a semi-infinite tube. The tube is analogous to an external line of a Feynman graph, and what would be inserted at the end of it is a quantum string state. The relation between the two pictures is the correspondence between operators and states.

To understand the Weyl transformation between the two pictures, consider the metric of the plane (figure 4) in polar coordinates:

ds*2=dr*2+r*2dθ*2.( equation no 19)

We think of inserting an operator at the point r = 0.

figure

[Figure 4. A plane ℝ2, when a labeled point p is omitted, is equivalent via a Weyl transformation to a cylinder with a flat metric. Vertical position on the cylinder is given by w and the point p is mapped to the bottom end of the cylinder at w = −∞.]

Now remove the point and make a Weyl transformation by multiplying ds2 with 1/r2 to get a new metric

(ds′)*2=1/r*2 dr*2+dϕ*2…….equation no 20)

In terms of w = log r, −∞ < w < ∞, the new metric is (ds′)*2=dw*2+dϕ*2,...(equation no 21) which describes a cylinder. The point r = 0 in one description corresponds in the other description to the w → −∞ end of the cylinder. What is interpreted in one description as an operator inserted at r = 0 is interpreted in the other description as a quantum state flowing in from w = −∞.

Thus to us , string theory can describe quantum gravity in spacetime. But it does not describe quantum gravity only. It describes quantum gravity unified with various particles and all forces in spacetime. The other particles and forces correspond to other operators in the conformal field theory of the string—apart from the operator 𝒪 that is related to a fluctuation in the spacetime geometry—or equivalently to other quantum states of the string.

The operator–state correspondence that leads to string theory describing quantum gravity in spacetime is also important in some areas of statistical mechanics and condensed-matter physics. That is indeed another one of nature’s rhymes.

No ultraviolet divergences

The next step for us is to explain why this type of theory does not have ultraviolet divergences?, in sharp contrast to what happens if we simply apply textbook recipes of quantization to the Einstein–Hilbert action for gravity. When we use those recipes, we encounter intractable ultraviolet divergences that were first found in the 1930s. Back then it was not entirely clear that the problem is special to gravity, because there were also troublesome ultraviolet divergences when other particle forces were studied in the framework of relativistic quantum theory. However, as ultraviolet divergences were overcome for the other forces—most completely with the emergence of the standard model of particle physics in the 1970s—it became clear that the problems for gravity are serious.

To understand why there are no ultraviolet divergences in string theory, we should begin by asking how ultraviolet divergences arise in ordinary quantum field theory. They arise when all the proper-time variables in a loop go simultaneously to zero. So in the example of figure 1, there can be an ultraviolet divergence when τ1, τ2, τ3, and τ4 simultaneously vanish.

It is true that a Riemann surface can be characterized by complex parameters that roughly parallel the proper-time parameters of a Feynman graph (figure 2). But one important difference prevents ultraviolet divergences in string theory. The proper-time variables τi of a Feynman graph cover the whole range 0 ≤ τi ≤ ∞. By contrast, the corresponding Riemann surface parameters τ̂i are bounded away from zero. Given a Feynman diagram, one can make a corresponding Riemann surface, but only if the magnitudes of the proper-time variables τ̂i are not too small. The region of the parameter space where ultraviolet divergences occur in field theory simply has no counterpart in string theory.

Instead of giving a general explanation, we will show how it works in the case of the one-loop cosmological constant. The Feynman diagram is a simple circle (figure 5a), with a single proper-time parameter τ. The resulting expression for the one-loop cosmological constant is

figure

[Figure 5. One-loop cosmological constant. (a) In quantum field theory, this Feynman diagram with a single proper-time parameter τ, underlies the one-loop cosmological constant. (b) The string theory counterpart is a torus characterized by a parameter u (the imaginary part of the complex parameter τ̂ from figure 2a) that, crucially, is bounded away from zero.]

Λ1=1\2∫*∞**0 dτ/τ Tr exp(−τH),(equation no 22) where H is the particle Hamiltonian ½(P*2 + m*2). The integral diverges at τ = 0, and the divergence is actually more severe than it looks because of the momentum integration that is part of the trace.

Going to string theory means replacing the classical one-loop diagram with its stringy counterpart, which is a torus (figure 5b). Nineteenth-century mathematicians showed that every torus is conformally equivalent to a parallelogram in the plane with opposite sides identified:

Figure

But to explain the idea without any extraneous technicalities, we will consider, instead of parallelograms, only rectangles:

Figure

We label the height and base of the rectangle as s and s′, respectively.

Only the ratio u = s’/s is conformally invariant. Also, since what we call the “height” as opposed to the “base” of a rectangle is arbitrary, we are free to exchange s and s′, which corresponds to u ↔ 1/u. So we can restrict ourselves to s′ ≥ s, and thus the range of u is 1 ≤ u < ∞.

The proper-time parameter τ of the particle corresponds to u in string theory, with the key difference being that for the particle, 0 ≤ τ < ∞, but for the string, 1 ≤ u < ∞. So in the approximation of considering only rectangles and not parallelograms, the one-loop cosmological constant in string theory is

Λ1=1/2∫*∞**1t d**u/u0Trexp(−uH)....(equation no 23)

There is no ultraviolet divergence, because the lower limit on the integral is 1 instead of 0. A more complete analysis with parallelograms shifts the lower bound on u from 1 to √3/2.

We have described a special case, but the conclusion is general. The stringy formulas generalize the field theory formulas, but without the region that can give ultraviolet divergences in field theory. The infrared region (τ → ∞ or u → ∞) lines up properly between field theory and string theory, and that is why a string theory can imitate field theory in its predictions for behavior at low energies or long times and distances.

Emergent spacetime

Our next attempt here will be to explain, in what sense spacetime emerges from something deeper if string theory is correct. Let us focus on the following fact. The spacetime M with its metric tensor GIJ(X) was encoded as the data that enabled us to define one particular 2D conformal field theory. That is the only way that spacetime entered the story.

In our equations, we could have used a different 2D conformal field theory. Now if GIJ(X) is slowly varying (the radius of curvature is everywhere large), the Lagrangian by which we described the 2D conformal field theory is weakly coupled and useful. In that case, string theory matches the ordinary physics that we are familiar with. In this situation, we can say that the theory has a semiclassical interpretation in terms of strings in spacetime—and it will reduce at low energies to an interpretation in terms of particles and fields in spacetime.

When we get away from a semiclassical, weak-coupling limit, the Lagrangian is not so useful and the theory does not have any particular interpretation in terms of strings in spacetime. The breakdown of a simple spacetime interpretation has many nonclassical consequences, such as the ability to make continuous transitions from one spacetime to another, or the fact that certain types of singularities (but not black hole singularities) in classical general relativity turn out to represent perfectly smooth and harmless situations in string theory. An example of the nonclassical behavior of string theory is sketched in figure 6.

figure

[Figure 6. Schematic representation of a family of two-dimensional conformal field theories (the gray region bounded by black lines) that depend on two parameters. For some values of the parameters, the theories have semiclassical interpretations in terms of strings propagating in a spacetime M1, M2, or M3. Generically there is no such interpretation yet present in string theory. However, one can make a continuous transition from one possible classical spacetime to another, as indicated by the colored lines.]

In general, string theory comes with no particular spacetime interpretation, but such an interpretation can emerge in a suitable limit, somewhat as classical mechanics sometimes arises as a limit of quantum mechanics. From this point of view, spacetime emerges from a seemingly more fundamental concept of 2D conformal field theory.

Wechave not given a complete explanation of the sense in which, in the context of string theory, how spacetime emerges from something deeper. A completely different side of the story, beyond the scope of the present article, involves quantum mechanics and the duality between gauge theory and gravity. (See the article by Igor Klebanov and Juan Maldacena, Physics Today, January 2009, page 28.) However, what we have described in this article is certainly one important and relatively well-understood piece of the puzzle. It is at least a partial insight about how spacetime as conceived by Einstein can emerge from something deeper, and thus hopefully is of interest in the present centennial year of general relativity.

figure

References

1)Rupak Bhattacharya, Dr Pranab Kumar Bhattacharya, Upasana Bhattacharya, Ritwik Bhattacharya, Rupsa Bhattacharya, Dalia Mukherjee, Ayushi mukherjee, Debasish mukherjee, soumayak Bhattacharya, Hindole Banerjee . "What if the Big Bang was not the bigining of our observable universe." Research and Review: journal of space science and technology STM journals Vol 10; issue no 3; page 1-7;2021

2)Murray GellMan " Factorization of couplings to Regge pols " Physical Review Letters vol 8,p 263,1962

3)Zweig G ( CERN) "An Su3 model for string interaction symmetry and it's breaking " CERN preprint 8182/ TH.401 January 17, 1964 and in article " Development of quark theory of Hadrons " pp 21-104

4) Close FE " Introduction of Quarks and protons " Academic press ,Waltham 1979 authors Bin Liang journal of modern physics vol 6, no 7 June 39 2015

5)Bhattacharya Rupak, Bhattacharya Ritwick, Mukherjee Dalia, Bhattacharya Pranab Kumar "Sub2 quark particle possible in the origin of mass in early universe " , discussion in Extreme Astronomy.com

6)Alexander Vilenkin Gravitational field of vacuum domain walls and strings

Phys. Rev. D 23, 852 – Published 15 February 1981

7) Brosche, P. Z. Astrophys. 57, 143–155 (1963). cited in article "Cosmic strings, superstrings and the evolution of the Universe" by author L.J. Tassie

Nature volume 323, page 40–42 1986

8)Rupak Bhattacharya, Pranab Kumar Bhattacharya, Upasana Bhattacharya, Ritwik Bhattacharya Rupsa Bhattacharya, Dalia Mukherjee etal 'Tachyon faster than light particles exist in our universe or an imaginary mathematical particles " International journals of Astronomy Astrophysics and Space Science vol 2; no 3; June 2015; page 12-29).

9)Alexander Vilenkin "Cosmological Density Fluctuations Produced by Vacuum Strings" Phys. Rev. Lett. 46, 1169 – Published 27 April 1981; Erratum Phys. Rev. Lett. 46, 1496 ;1981

10) L. J. Tassie "Cosmic strings, superstrings and the evolution of the Universe" Nature volume 323, pages 40–42; 1986

11) Bahcall, N. A., & Soneira, R. M. 1983, ApJ, 270, 20 [G

12) Reference No2 P J E Peebles as published articles" Large-scale background temperature and mass fluctuations due to scale-invariant primeval perturbations" Astrophysical Journal, Part 2 - Letters to the Editor, vol. 263, Dec. 1, 1982, p. L1-L5. In the article Formation of galaxies and large-scale structure with cold dark matter by authors George R. Blumenthal, S. M. Faber, …Martin J. Rees

Nature volume 311, pages 517–525 (1984)

13) Edward written String theory Physics today November 2015

Extra references for studying about string theory

1 B. Zwiebach, A First Course in String Theory, 2nd ed., Cambridge U. Press (2009).

2 J. Polchinski, String Theory, Volume 1: An Introduction to the Bosonic String, Cambridge U. Press (2005).

3 M. B. Green, J. S. Schwarz, E. Witten, Superstring Theory, Volume 1: Introduction, Cambridge U. Press (1987).

4=Highlighting the usefulness of string theory

5 String Theory and M-Theory: A Modern Introduction

6 String Theory and Particle Physics: An Introduction to String Phenomenology

Is string theory phenomenologically viable?

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