Limits of Relativistic Systems Bachelor Degree Project

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Limits of Relativistic Systems

Bachelor Degree Project

Marcus St˚ alhammar

Supervisor:

Ulf Lindstr¨ om

Institutionen f¨or fysik och astronomi, Uppsala Universitet, Box 803, SE-751 08 Uppsala, Sweden

E-mail: marcus.stalhammar.9957@student.uu.se Abstract

We investigate the massless and tensionless limit for the relativistic particle and string respectively. The interaction with interesting background fields, such as the electromagnetic field in the particle case and the Kalb-Ramond field in the string case, are also studied. In order to take the massless- and tensionless limit of the corresponding action, a Hamiltonian field theory is used. The invariances under the Poincar´e group and diffeomorphisms are of great use, and the Weyl invariance in the string case is found particularly useful, since the background metric can be chosen to be flat without loss of generality. We find it useful to study these systems in the transversal gauge, which is the tensionless correspondence to the conformal gauge, in which the action attains a convenient form. Furthermore, we discuss the physical meaning of some of the results, identify some possible further investigations and also some applications in other fields of physics.

Sammanfattning

Vi undersöker den masslösa respektive spänningslösa gränsen för en relativistisk partikel och sträng, b˚ade i det fria fallet och under inverkan av elektromagnetisk bakgrund. Genom att använda en Hamiltoniansk fältteori och utnyttja dess invarians under Poincarétransformationer och diffeomorfismer, kan dessa gränser studeras. För att förenkla strängfallet, används Weylinvarians för att kunna betrakta fallet med en platt bakgrundsmetrik, utan att teorin blir mindre generell. En spänningslös analogi av den konforma gaugen, den transversella gaugen, har visat sig vara användbar för att verkan för strängen skall kunna skrivas p˚a en enkel form. Slutligen diskuteras resultatens fysiska mening, applikationer inom andra fysiska teorier och hur teorin skall vidareutvecklas.

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1 Introduction

Since the 1960s, string theory has been a popular field of research within the vast subject of Physics. At the beginning, it was developed in hope to construct a theory describing the strong interaction between protons and neutrons, and between the quarks within them.

Due to some calculations problems, string theory was abandoned for a while, just to be picked up again. Then its true potential was discovered. Firstly, almost all of Einstein’s general theory of relativity is contained within the framework of string theory, and therefore it predicts gravity. This is in contrast to other famous theories, e.g. quantum field theory, which do not allow the existence of gravity. Moreover, it is the only theory so far containing a quantum theory of gravitation. String theory has the potential to provide a complete description of both particle physics and cosmology. ¹

Like in most parts of physics, specific critical limits might be of a certain interest. Here the tensionless limit of the relativistic string serves as an excellent example. For the sake of clarity, we will introduce some fundamental concepts of strings here. A string can either be open or closed. A closed string is considered to be small loop, spanning a pipe-like world sheet when traveling through space. An open string, which will be considered throughout this thesis, really looks like a string, with two endpoints. The tension of a string is a generalized concept of mass with units of energy per unit length. One of the interesting things about this limit is that one can use the same mathematical method, only a bit generalized, as for the massless particle. Even though the calculations are very similar, new interesting results arise from the string case.

2 Background

Studies on tensionless string theory have been done since the 1980s and there is still ongoing research. In the early 1990s, the authors of [2] investigated the tensionless limit of the relativistic superstring and derived its field equations. During the following years, improvements and generalizations where made, finally arriving at [3]. Almost exclusively, a Hamiltonian field theory was used in order to receive a non-trivial description of the tensionless limit.² Recent publications, as [4, 5], show that applications in other fields of physics are possible, and also that there still is a lot to be found.

1This section is based on the discussion given in [1]

2The basic concepts of Lagrangian and Hamiltonian field theory will be given in the theory section.

The specific formalism and techniques used will be presented along with the results. Detailed description will be given in the appendix.

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3 Theory

In order to facilitate reading and motivate some of the method used, we will give a short presentation of some mathematical tools and how to apply them in physics. Throughout this section, Latin indices will correspond to space indices, and Greek indices will correspond to Minkowski indices. The various examples discussed are based on and inspired by examples discussed in [6–8].

3.1 Calculus of Variation

Below we give a mathematical derivation of the Euler-Lagrange equations, which are central in analytical mechanics and for classical field theory. The variational principle used is called the fixed endpoint problem and has important physical meaning, which will be discussed briefly below.

To begin with, we define the functional, J =

Z

f (x, y, y⁰) dx, (3.1)

where y = y(x) and y⁰ = ^dy_dx. Furthermore, define the following two sets,

S = {y ∈ C²[x₀, x₁] : y(x₀) = y₀, y(x₁) = y₁}, (3.2) H = {η ∈ C²[x0, x1] : η(x0) = η(x1) = 0}, (3.3) where C²[x₀, x₁] is the vector space of all two-times continuously differentiable functions on the interval [x₀, x₁], equipped with the norm ||.||. A small change in the function y may be written as ˆy = y + η for any real number > 0. We recall the definition of local extrema. J is said to have a local maxima in S if

∃ > 0 : J(ˆy) − J (y) ≤ 0 ∀ˆy ∈ S : ||ˆy − y|| < . (3.4) Likewise J has a local minimum if −J has a local maximum.

In order to derive the Euler-Lagrange equations, we need to be familiar with the definition of the first variation. With the help of Taylors theorem, we know that we can write f (x, ˆy, ˆy⁰) as an expansion in the following manner,

f (x, ˆy, ˆy⁰) = f (x, y, y⁰) +

η∂f

∂y + η⁰∂f

∂y⁰

+ O(²), (3.5)

and hence we can investigate small deviations from J , and by linearity of the integral, we find,

J (ˆy) − J (y) = Z x1

x0

η∂f

∂y + η⁰∂f

∂y⁰

dx + O(²) ≡ δJ (η, y) + O(²). (3.6) The term δJ (η, y) is what we define as the first variation of the functional J . We clearly see that if η ∈ H, then −η ∈ H, which gives us δJ (η, y) = −δJ (−η, y). For small

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enough, the sign of (3.6) is determined by δJ , except for vanishing first variations, i.e.

δJ = 0 ∀η ∈ H. But if J(y) is a local maximum in S, then (3.6) shall not change sign for any ˆy ∈ S such that ||ˆy − y|| < , and then δJ (η, y) must vanish for all η ∈ H. The same reasoning gives the same result if J (y) is a local minimum.

If the first variation vanishes, then the functional J is said to be stationary at y. For a stationary functional, we can conclude, using partial integration and the definition of η,

0 = Z x1

x0

η∂f

∂y + η⁰∂f

∂y⁰

dx =

= Z x1

x0

η∂f

∂y − η d dx

∂f

∂y⁰

dx +

η∂f

∂y⁰

x1

x0

⇒ 0 = ∂f

∂y − d dx

∂f

∂y⁰

, (3.7)

which is the result we wanted! ³

As mentioned above, the fixed endpoint problem is of great interest in physics. Below, the rest will be generalized and derived for continuous systems. Also, the physical im- portance will be demonstrated. One thing worth mentioning now already, is the fact that the Euler-Lagrange equations are invariant under coordinate changes, which can easily be shown by using the Chain rule repeatedly.

3.2 Extension to Continuous Systems and Field Theories

In classical- and basic analytical mechanics, one deals almost exclusively with discrete mechanical systems possessing a finite number of degrees of freedom ϕⁱ. Many physical systems are not discrete, but continuous, and they are described by fields. As examples, we have the electromagnetic field and continuous matter. These systems can be described in a similar fashion as the discrete ones, and we will show this by studying what happens with a discrete system described by ϕⁱ when we let the index i become a continuous index x. We will label the fields, which in our case will be scalar fields, by ϕ (x, t) = ϕ(t).

3.2.1 The Lagrangian Density

For later use, we begin by introducing the concept of the functional derivative of a functional with respect to a field. For an arbitrary functional F [ϕ], we have,

δF [ϕ] ≡ F [ϕ + δϕ] − F [ϕ] =

Z δF [ϕ]

δϕ δϕd³x, (3.8)

where ^{δF [ϕ]}_δϕ is the functional derivative with respect to ϕ at a specific point x. Extend- ing this to a functional of several functions is a straight forward generalization from the corresponding reasoning for usual derivatives.

3Even though this a derivation for the one dimensional case, the general case is a quite straightforward generalization. This derivation will be given below, in a four-vector formalism.

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Consider a continuous system described by the Lagrangian L (ϕ, ˙ϕ, t), which is a functional of the fields ϕ and ˙ϕ, which are functions of x. The variation of the Lagrangian will look like, in accordance to extending (3.8),

δL (ϕ(t), ˙ϕ(t), t) = Z

d³x

δL

δϕ(t)δϕ(t) + δL

δ ˙ϕ(t)δ ˙ϕ(t)

. (3.9)

If we instead consider the Lagrangian to be a functional of the space-dependence of ϕ and ˙ϕ, it is natural to think of it as a function of infinitely many discrete variables in the continuous limit, namely,

L (ϕi(t), ˙ϕi(t), t) −−→

i→x L

ϕ(t),ϕ(t), t˙

. (3.10)

Now, let the values of ϕ at every point x be canonical (independent) coordinates.

Consider the usual three-space, and divide it into tiny cells of volume δVⁱ. Suppose that every cell is described by,

L(t) = L (ϕ_i(t), ˙ϕ_i(t), t) , (3.11) where ϕ_i(t) is the mean value of ϕ(x, t) in cell number i. L(t) is then a functional of ϕ_i and ˙ϕ_i in every cell. Thus, the variation attains the following form,

δL (ϕ_i(t), ˙ϕ_i(t), t) =X

i

∂L

∂ϕ_iδϕ_i+ ∂L

∂ ˙ϕ_iδ ˙ϕ_i

=X

i

1 δVⁱ

∂L

∂ϕ_iδϕ_i+ ∂L

∂ ˙ϕ_iδ ˙ϕ_i

δVⁱ. (3.12)

Identifying (3.12) with (3.9), we see that the functional derivative is a continuous limit, i.e. δVⁱ → 0, of the ordinary derivative,

δL

δϕ ≡ lim

δVⁱ→0

1 δVⁱ

∂L

∂ϕ_i (3.13)

δL

δ ˙ϕ ≡ lim

δVⁱ→0

1 δVⁱ

∂L

∂ ˙ϕ_i, (3.14)

here it is important that x ∈ δVⁱ. We can regard δVⁱ ≡ d³x in this limit.

Since we know that the Euler-Lagrange equations hold in the discrete case (due to Fixed endpoint principle),

∂

∂ϕ_i − ∂t

∂L

∂ ˙ϕ_i = 0, (3.15)

we get the following in the continuous limit, δL

δϕ − ∂_tδL

δ ˙ϕ = 0. (3.16)

Moving on, we assume the the total Lagrangian L can be written as a sum of L_i, defined in the i^th cell for every i,

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L (ϕ_i, ˙ϕ_i, t) =X

i

L_i(ϕ_i, ∇ϕ_i, ˙ϕ_i, t) , (3.17) where Li is a functional of the fields ϕi, the gradient of the fields ∇ϕi and the time derivatives ˙ϕ_i. The expression (3.17) is very suitable for taking the continuous limit, at least after multiplying and dividing with δVⁱ,

L (ϕ_i, ˙ϕ_i, t) =X

i

L_i(ϕ_i, ∇ϕ_i, ˙ϕ_i, t) −−→

δVⁱ (3.18)

→ Z

V

d³xL (ϕ(x), ∇ϕ(x), ˙ϕ(x); x, t) , (3.19) with the identifications x ≡ (x^µ) and the Lagrangian density L ≡ lim_δVⁱ_→0 _δV¹iL_i(ϕ, ∇ϕ, ˙ϕ, t).

In terms of the Lagrangian density, the action is defined as, S =

Z t2

t1

L(t)dt = Z

d⁴xL(x), (3.20)

where the last integral is taken over a domain [t₁, t₂] × V ⊆ M₄, where M₄ is Minkowski space. Under Lorentz transformation, the action should transform as a scalar quantity, and hence, L must transform as a scalar. Thus, L must depend on both space- and time derivatives of ϕ, and the dependence must be the same, i.e., L depends on the invariants constructed out of ∂_µϕ.

Consider infinitesimal variations of ϕ, which vanishes on the boundary of the domain of integration. In analogy to (3.12), the variation of the total Lagrangian is,

δL = Z

d³x L

∂ϕδϕ + L

∂∂_iδ∂i+∂L

∂ ˙ϕδ ˙ϕ

= (3.21)

= Z

d³x L

∂ϕ − ∂i

L

∂∂_iϕ

δϕ − ∂L

∂ ˙ϕδ ˙ϕ

, (3.22)

where we identify the integrand as the functional derivatives,

δL

δϕ = ∂L

∂ϕ − ∂i

∂L

∂∂_iϕ

(3.23) δL

δ ˙ϕ = ∂L

∂ ˙ϕ, (3.24)

and the Euler-Lagrange equations in terms of the Lagrangian density becomes,

∂L

∂ϕ − ∂_µ

∂L

∂ (∂_µϕ)

= 0. (3.25)

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3.2.2 The Hamiltonian Density

In the discrete case, we can define a momentum p_i conjugate to the fields ϕ_i by, p_i = ∂L(t)

∂ ˙ϕ(t), (3.26)

and the Hamiltonian H is defined by the Legendre transformation,

H =X

i

(piϕ˙i− L) , (3.27)

with the equations of motion defined by,

˙

ϕ_i = ∂H

∂p_i, (3.28)

˙

p_i = −∂H

∂ϕ_i. (3.29)

In parallell to the case of the Lagrangian, we want to investigate the continuous limit here as well. By the same reasoning, i.e. dividing the space into small cells of volume δVⁱ, we conclude,

H = X

i

δVⁱ(p_iϕ˙_i− L) 1

δVⁱ =X

δVⁱ ∂L

∂ϕⁱϕ˙_i− L

1

δVⁱ −−−−→

δVⁱ→0

→ Z

d³x ∂L

∂ϕϕ − L˙

≡

≡ Z

d³x (π ˙ϕ − L) ≡ Z

d³xH, (3.30)

where π ≡ ^∂L_∂ϕ is the momentum density canonical to ϕ and H is the Hamiltonian density.

Hamiltons field equations are easily derived from a variational principle, using that the action should be stationary and that the variations of the fields δϕ should vanish at the initial and final time. We get,

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δS = Z tf

ti

dt Z

d³xδ (π ˙ϕ − H) =

= Z tf

ti

dt Z

d³x

˙

ϕδπ + πδ ˙ϕ − ∂H

∂ϕδϕ +∂H

∂πδπ

=

= Z tf

ti

dt Z

d³x

˙

ϕδπ + π∂₀δϕ − ∂H

∂ϕδϕ + ∂H

∂πδπ

=

= Z tf

ti

dt Z

d³x

δπ

˙ ϕ − ∂H

∂π

− δϕ

˙π + ∂H

∂ϕ

+ Z

d³x δϕπ|^t_t^f_i =

= Z tf

ti

dt Z

d³x

δπ

˙ ϕ − ∂H

∂π

− δϕ

˙π + ∂H

∂ϕ

= 0 ⇒ (3.31)

⇒ ˙ϕ = ∂H

∂π, (3.32)

˙π = −∂H

∂ϕ, (3.33)

where we have assumed ^∂H_∂t = 0.

One sees that the continuous case is very similar to the discrete case. Throughout the rest of the thesis, we might loosely speak of the Lagrangian density and Hamiltonian density as the Lagrangian and the Hamiltonian. It should be clear though that we will use a continuous theory exclusively.

4 Relativistic Particle - a Mathematical Approach

4.1 Equations of Motion for a Free Particle

The action for a free point particle is written as, S = −

Z

M ds, (4.1)

but it is often convenient to consider it on the following form, S = −M

Z p

− ˙X²dt. (4.2)

Here, M is the mass of the particle, ˙X^µis the time derivative of the generalized coordinate X^µ and ds² ≡ −dX², where dX² = dX_µdX^µ. Also, X_µX^µ= X².

We identify the Lagrangian density as, L = −M

q

− ˙X², (4.3)

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from which we can define the momentum conjugate to X^µ, using the usual equations from analytical mechanics,

P_µ= ∂L

∂ ˙X^µ = −M − ˙X^µ p− ˙X²

. (4.4)

By squaring the the momentum, we achieve the following constraint, ⁴

P²+ M² = 0, (4.5)

using the same conventions as above, PµP^µ= P².

Now we want to pass to phase space, and this we do by the usual Legendre transformation, H₀ = P_µX˙^µ− L, but the naive Hamiltonian defined like this vanishes. This is due to the invariance under diffeomorphisms, i.e. smooth coordinate changes. Using Lagrange multipliers, the Hamiltonian attains the following form,

H = λ P²+ M² , (4.6)

with λ a Lagrange multiplier.

In order to reach our goal, which is an expression that allows us to take the massless limit, we introduce the phase space action,⁵

S^{P S} = Z

P_µX˙^µ− λ P²+ M² dt. (4.7) Using Hamilton’s canonical equation, ˙X^µ = _∂P^∂H

µ and solving for P^µ, the momentum can be written as

P^µ= X˙^µ

2λ. (4.8)

Using (4.8) to integrate out the momentum, one passes to configuration space and the following form of the action,

S^CS =

Z X˙²

4λ − λM²

!

dt. (4.9)

This clearly is suitable for taking the massless limit, and the equations of motion follow from a variational principle, or directly from the Euler-Lagrange equations. Using a variational principle (Fixed endpoint problem), one finds equations of motion in both phase- and configuration space as follows:

4A constraint is a relation between the generalized coordinates and canonical momentum independent of time, which is always satisfied, regardless of the motion of the system.

5Note that this expression still is an integral of the Lagrangian, i.e. it is equivalent to the form (4.2).

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





P˙_µ = 0, P²+ M² = 0, X˙^µ = 2P_µλ,

(4.10)

( ¨X^µλ − ˙X^µ˙λ = 0,

X˙²+ (2λM )² = 0. (4.11)

4.2 Equations of Motion for a Charged Particle

The procedure here will be very similar to what we saw in the previous subsection. The contribution from the electromagnetic field will change the action though, which will be,

S = − Z

Mp

− ˙X²dt − Z

dX_µeA^µ= − Z

Mp

− ˙X²+ e ˙X_µA^µdt, (4.12) and therefore the Lagrangian is,

L = −Mp

− ˙X²− e ˙X_µA^µ. (4.13) As before, we define the momentum conjugate to X^µ as,

P_µ = ∂L

∂ ˙X^µ = +M X˙_µ p− ˙X²

− eA_µ, (4.14)

and the constraint in this case is,

(Pµ+ eAµ) (P^µ+ eA^µ) + M² = 0. (4.15) Again, the naive Hamiltonian will vanish, since,

H₀ = P_µX˙^µ− L = M X˙_µ p− ˙X²

− eA_µ

!

X˙^µ+ Mp

− ˙X²+ e ˙X_µA^µ = (4.16)

= M +

X˙² p− ˙X²

+p

− ˙X²

!

= 0 ⇒ (4.17)

H₀ = 0, (4.18)

and therefore, the final Hamiltonian can be written as,

H = λ(P_µ+ eA_µ) (P^µ+ eA^µ) + M² , (4.19) again with λ a Lagrange multiplier. Deriving a new expression for the momentum conjugate to X^µ, in analogy to the previous section, by using the Canonical equations, we get,

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P^µ= X˙^µ

2λ − eA^µ. (4.20)

Defining the phase space action, S^{P S} =

Z h

P_µX˙^µ− λ(Pµ+ eA_µ) (P^µ+ eA^µ) + M²i

dt, (4.21)

and integrating out the momentum, again yields an action suitable for the massless limit, namely,

Z X˙²

4λ − eA_µX˙^µ− λM²

!

dt. (4.22)

Again, the equations of motion will be reached either by using the Euler-Lagrange equations or by using the same variational principle as before. By doing the latter, one obtains the equations of motion in both phase- and configuration space.







X˙^µ− 2λ (P^µ+ eA^µ) = 0 (P_µ+ eA_µ) (P^µ+ eA^µ) + M² = 0 P˙_µ+ 2eλ∂_µA_ν(P^ν + eA^ν) = 0

(4.23)

(X˙² + (2M λ)² = 0

X¨^µλ − ˙X^µ˙λ = 2λ²e ˙X^ν(∂_νA^µ− ∂^µA_ν) (4.24)

5 Relativistic String - a Mathematical Approach

The generalization from a particle to a string is quite straight forward. When deriving the equations of motion for the particle, we took the length of the world-line as the action. In the string case, we will simiralrily use the area spanned by the ends of the string, i.e. the area of the world-sheet, as the action. For more general object, i.e. p-branes⁶, we similarly take the size of the world-volume as the action.

5.1 Equations of Motion for Free String

In the case of a p-brane, we have the following action and Lagrangian, S = T

Z

d^p+1ξp− det γ_ij ≡ Z

d^p+1ξL, (5.1)

6A p-brane is a generalized version of the particle and the string, with extension in p dimensions. Thus, it sweeps out a p + 1 dimensional world-volume when traveling through Minkowski space.

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with the identifications,

γ_ij ≡ ∂_iX^µ∂_jX^νη_µν, (5.2)

γ_ijγ^kl ≡ δ_i^kδ_j^l, (5.3)

det (γ_ij) ≡ γ, (5.4)

where i = 0, 1, ..., p and η_µνis the Minkowski metric. We will use the signature (−, +, +, +).

The conjugate momentum defined from (5.1) is, Pµ = T√

−γγⁱ⁰∂iXµ, (5.5)

from which we can define the naive Hamiltonian density, using the usual Legendre transformation H = PµX˙^µ − L, which vanishes due to the diffeomorphism invariance of the theory. As in the case of the particle, we can derive constraints, but here we will get two different expressions. One is received by multiplying (5.5) by ∂_aX^µ, a = 1, ..., p from the right, and the other one by multiplying (5.5) by P^µ. We get,

P²+ γT²γ⁰⁰ = 0, (5.6)

P_µ∂_aX^µ = 0, (5.7)

with P_µη^µνP_ν = P². Using them, we can write the Hamiltonian density as:

H = λ P²+ γT²γ⁰⁰ + ρ^aPµ∂aX^µ, a = 1, ..., p, (5.8) from which we can define another expression for the momenta by inverting ˙X^µ= _∂P^∂H

µ and

thus receiving,

P^µ=

X˙^µ− ρ^a∂_aX^µ

2λ . (5.9)

Now we can introduce a phase space action, S^{P S} =

Z h

PµX˙^µ− λ P²+ T²γγ⁰⁰ − ρ^aPµ∂aX^µ i

d^p+1ξ, (5.10) and integrating out the momenta yields,

S^CS =

Z (

X˙_µ− ρ^a∂_aX_µ 2λ

X˙^µ− λ

"

X˙_µ− ρ^a∂_aX_µ 2λ

! X˙_µ− ρ^a∂_aX_µ 2λ

!

+ T²γγ⁰⁰

#

−

−ρ^aX˙_µ− ρ^b∂_bX_µ 2λ ∂_aX^µ

)

d^p+1ξ = (5.11)

=

Z 1 4λ

hX˙²− 2ρ^aX˙_µ∂_aX^µ+ ρ^aρ^b∂_aX^µ∂_bX_µ− 4λ²T²γγ⁰⁰i

d^p+1ξ. (5.12)

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The action of the form (5.12) is non-vanishing in the tensionless limit. By introducing Vⁱ = ¹

2√

λ(1, ρⁱ) ⁷, the action in the string case (p = 1) attains the following form, S^N =

Z

VⁱV^jγ_ijd²ξ, (5.13)

and using the Euler-Lagrange equations gives us the following field equations.

∂_i VⁱV^j∂_jX^µ

= 0, (5.14)

V^jγij = 0. (5.15)

Using (5.14) and (5.15) together, it is clear that the metric tensor has a null eigenvector, and the equations reduce to,







γ = 0,

X¨^µ = 0,

X˙² = ˙XX⁰ = 0,

(5.16)

which tells us that the metric tensor γ_ij is degenerate and thus the string spans a null- surface in Minkowski space. Moreover, for an open string, we have a boundary conditions which need to be satisfied to have the correct field equations. Since there are several choices here with different physical meaning, we will deal with this in the discussion part.

5.2 Equations of Motion for a String in the Presence of an Anti-

Symmetric Field

The reasoning of this section is all based on [3]. In the case of an open string, or p-brane, interacting with an anti-symmetric background field, the action will look like,

S = T Z q

− det (γ_ij+ F_ij)d^p+1ξ ≡ Z

Ld^p+1ξ, (5.17)

where we have the identifications,

γij ≡ ∂iX^µ∂jX^νGµν, (5.18)

F_ij ≡ ∂_iA_j − ∂_jA_i+ B_ij ≡ ∂_[iA_j]+ B_ij, (5.19)

B_ij ≡ ∂_iX^µ∂_jX^νB_µν, (5.20)

7Vⁱ here transforms as a vector density, i.e. it transforms as a vector up to a power of the Jacobian determinant corresponding to the coordinate change. The specific power is called the weight of the vector density, which in this case will be ¹₄.

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where the gauge field A lives only on the boundary of the p-brane. B_µν is the Kalb-Ramond field.

Since we have more than one field apparent in the expression of the action, we will have several generalized momenta in this case, one conjugate to each component of the two fields. The definition of the momenta is equivalent to what it have been before, thus we have,

P^a = ∂L

∂ ˙A_a = T

2p− (γ + F ) (γ + F )⁻¹^[a0]

, (5.21)

P⁰ = ∂L

∂ ˙A₀ = 0 since F00= 0, (5.22)

Π_µ = ∂L

∂ ˙X^µ = T

2p− (γ + F ) (γ + F )⁻¹(i0)

∂_iX^µ. (5.23) As before, we will have some constraints here as well. The calculations are rather lengthy and highly non-trivial, and the reader may look them up in the appendix. We will use the following relations,

(γ + F )⁻¹^(ik)

F_kl+(γ + F )⁻¹^[ik]

γ_kl = 0, (5.24)

(γ + F )⁻¹^(ik)

γ_kl+(γ + F )⁻¹^[ik]

F_kl = 2δ_lⁱ, (5.25) which will be derived in the appendix as well, and the primary constraints will be,⁸

Πµ∂aX^µ+ P^bFab = 0, (5.26) Π_µΠ^µ+ P^aγ_abP^b+ T²det ((γ + F )_ab) = 0, (5.27)

P⁰ = 0. (5.28)

Finding the field equations for A₀ yields a secondary constraint (which can also be found due to the preservation of the primary constraints),

∂_iPⁱ = 0. (5.29)

In this case, the theory is not diffeomorphism invariant. Thus, the naive Hamiltonian density will not vanish completely, but the constraints can still be added in the same manner as before which results in,

H = P^a∂_aA₀+ σP⁰+ ρ^a Π_µ∂_aX^µ+ P^bF_ab +

+λΠ_µΠ^µ+ P^aγ_abP^b+ T²det ((γ + F )_ab) , (5.30) were P^kA˙_k = 0 since k is not a transversal index. Here, λ, σ and ρ^a are Lagrange multipliers. Using (5.30) we get the following expression for the momenta,

8A primary constraint is always satisfied and contains relations between the momenta and coordinates, without explicit time dependence. A secondary constraint is satisfied when the equations of motions are satisfied. They appear because of the dynamical preservation of the primary constraints. [9]

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X˙^µ = ∂H

∂Π_µ = 2λΠ^µ+ ρ^a∂_aX^µ⇒ Π^µ =

X˙^µ− ρ^a∂aX^µ

2λ (5.31)

A˙a = ∂H

∂P^a = ∂iA0δia+ ρ^cFcbδba+ λ γcbP^bδca+ P^cγcbδcb =

= ∂aA0+ ρ^cFca+ 2λP^bγba ⇒ P^b = F_0a− ρ^cF_ca

2λ γ^ba, (5.32)

and the phase space action is defined as, S^{P S} =

Z

d^p+1ξn

Π_µX˙^µ+ P^aF_0a− σP⁰− ρ^a Π_µ∂_aX^µ+ P^bF_ab −

−λΠ_µΠ^µ+ P^aγ_abP^b+ T²det ((γ + F )_ab) . (5.33) As before, we pass to configuration space by integrating out the momenta,

S^CS = Z

d^p+1ξ

(X˙_µ− ρ^a∂_aX_µ 2λ

X˙^µ+F_0c− ρ^aF_ac

2λ γ^caF_0a−

−λ

"

X˙µ− ρ^a∂aXµ

2λ

X˙^µ− ρ^a∂aX^µ

2λ +

+F_0c− ρ^aF_ac

2λ γ^caγ_abF_0c− ρ^bF_bc

2λ γ^cb+ T²det ((γ + F )_ab)

−

−ρ^a X˙_µ− ρ^b∂_bX_µ

2λ ∂_aX^µ+ F_0c− ρ^dF_dc 2λ γ^cdF_bd

!)

, (5.34)

and after some really lengthy and cumbersome calculations, we land in,

S^CS = Z

d^p+1ξ 1

4λγ₀₀− 2ρâγ_0a+ ρâρ^bγ_ab+ γâb(F_0a− ρ^cF_ca) F_0b− ρ^dF_db −

−4λ²T²(det (γ + F )_ab) . (5.35)

Our next step will be linearizing this expression. As always, details are given in the appendix. We introduce some additional Lagrange multipliers, θâ, σâb and Gâ, and moreover, we use a little mathematical trick. We will multiply expression (5.35) by (2 − 2 + 1), write all terms out explicitly, and then identify the Lagrange multipliers in a suitable way.

The identifications will be,

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γ^ab(F0a− ρ^cFca) F0b− ρ^dFdb (1 + 2 − 2) =

= γ^aeγefγ^{f b}(F0a− ρ^cFca) F0b− ρ^dFdb (1 + 2 − 2) =

= GêγefG^f + 2θâ(F0a− ρ^cFca) − 2θêγefG^f, and the action will attain the form,

S =

Z

d^p+1ξ 1

4λγ₀₀− 2ρâγ_0a+ ρâρ^bγ_ab+ Gâγ_abG^b+ 2θâ(F_0a− ρ^cF_ca) − γ_abG^b

−4λ²T²det (Hab) + σ^ab(H_ab− F_ab− γ_ab) . (5.36) Here H_ab ≡ F_ab+ γ_ab. By finding the equations of motion for H_ab and G^a, we integrate them out, which can be done since they are non-dynamical, and the action can then be written as,

S =

Z

d^p+1ξ 1

4λγ₀₀− ρâγ_0a+ θâF_0a− ρâγ_0a− θâF_0a+ ρâρ^b− θâθ^b+ 4λ²T²σâb γ_ab+ + θâρ^b− θ^bρâ+ 4λâT²σâb Fab+ 4λ²T²det (H_ab) (p − 1) . (5.37) Now the action is well suited for taking the tensionless limit. As in the previous section, we can facilitate the expression further by introducing vector densities, namely,

Vⁱ ≡ 1

√

λ(1, − (θ^a+ ρ^a)) , (5.38) Wⁱ ≡ 1

√λ(1, (θ^a− ρ^a)) , (5.39)

allowing us to write the action as, S = 1

4 Z

d^p+1ξVⁱW^j(γ_ij + F_ij) . (5.40) The field equations follow from Euler-Lagrange equations, and the result will be,

0 = W^j(γ_ij + F_ij) , (5.41)

0 = V^j(γ_ij + F_ij) , (5.42)

0 = ∂_j V^[jW^l] , (5.43)

0 = ∂_j V^(jW^l)∂_lX^µ . (5.44) As in the last case, we clearly see that the tensor (γ_ij + F_ij) has null-eigenvectors and thus that it is degenerate and has zero determinant. The difference here is that this does not imply that the world-volume is a null-surface. Further investigations will not be done within this thesis.

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6 Discussion

6.1 Interpreting Lagrange Multipliers - Particle Case

Considering again the massless relativistic particle, it might be interesting to investigate the actual significance of the Lagrange multipliers. Using the form (4.9) of the action, and by using the following identification,

e ≡ 2λ, (6.1)

the Lagrangian density attain the form, L = X˙²

2e − eM²

2 , (6.2)

and the action in the massless limit will look like, S₀ = 1

2 Z

e⁻¹X˙². (6.3)

The equations of motion for e, which is an auxiliary field, called the einbein field in the massive case, can easily be found by a variation of e,

e² = − X˙²

M². (6.4)

By integrating out e, we get,

S =

Z X˙²

2e − eM²

2 =

Z M ˙X² 2p

− ˙X²

− M 2

p− ˙X²

= −M Z p

− ˙X², (6.5)

which is what we started with.

Instead of describing e in terms of its equations of motion, we could parametrize it, i.e.

choose a specific gauge, as we did for the introduced vector densities in the string case. We will here look at the massless case exclusively. By choosing the gauge e = 1, the equations of motion for the free relativistic particle becomes,

( ¨X^µ = 0,

X˙² = 0. (6.6)

This is precisely the field equations obtained for the relativistic string, only projected to the particle case! The gauge chosen in the string case is called the transversal gauge, is equivalent to the gauge chosen here as well. It is clear, since ˙X² = 0, that the world-line of the massless relativistic particle is light-like, i.e., the particle is moving with the speed of

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light, which one would expect. How to interpret this in the string case will be dealt with below.

For the charged particle, the equations of motions in the same gauge will be (not to confuse the charge e in these expressions with the einbein above),

(X˙² = 0,

X¨^µ = e ˙X^ν(∂_νA^µ− ∂^µA_ν) , (6.7) which tells us that the world-line for the charged, massless relativistic particle will be light- like as well. So far, no such particles have been observed, but when observed, it will be known that it travels at the speed of light. The latter of the two equations corresponds to the Lorentz force. We will not discuss this matter further.

6.2 Interpreting Lagrange Multipliers - String Case

In the string case, we identified the Lagrange multipliers with the vector densities Vⁱ ≡

1 2√

λ(1, ρⁱ) in the tensionless limit. Before letting T → 0, it might be interesting to make another identification, namely,

g^ij =−1 ρ ρ −ρ²+ 4λ²T²

. (6.8)

Using this to re-write the action of the form (5.12), we get (for the string case)⁹, S^CS = −1

2T Z

d²ξ√

−gg^ijγij. (6.9)

This form of the action is of great interest since it possesses another symmetry. It is invariant under Weyl transformations, i.e. transformations of the form g_ij → Ωg_ij, where Ω is a scalar. The action is invariant since √

g → Ω√

g and g^ij → Ω⁻¹g^ij, which gives us,

S^CS = −1 2T

Z

d²ξ√

−gg^ijγij →

→ −1 2T

Z

d²ξΩ√

−gg^ijΩ⁻¹γ_ij =

= −1 2T

Z

d²ξ√

−gg^ijγ_ij. (6.10)

If p 6= 1, this invariance will not hold, since the determinant of the metric tensor will transform with a different power of Ω.

9One can make similar identifications in the case p > 1 as well, according to [10]. This is more complicated though, and beyond the scope of this project.

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The fact that the string action possesses this invariance is very useful. It allows us to consider a locally flat metric as globally flat. Therefore, we can pick our background metric to be flat, without loss of generality. The actual proof uses the notion of Ricci scalars of metrics and the fact that vanishing Ricci scalars imply flat metrics. Due to the lack of concepts presented in this thesis, the mathematical proof will not be given, but it can be found in, i.e. [11].

6.3 Boundary Conditions in the String Case

In string theory, it is well known that, for an open string, we have two different kinds of boundary conditions in order for the equations of motion to make sense. These are the Neumann- and the Dirichlet conditions.¹⁰ In the tensionless case, the usual expressions for the boundary conditions must be modified. The actual appearance of them would have been clear if the field equations would have been derived using the variation of the action (5.13). Hence, we will give that derivation here. The world sheet coordinates will be ξⁱ = (τ, σ). we let τ_i ≤ τ ≤ τ_f and 0 ≤ σ ≤ 1. The variation will look like,

δS = Z τf

τi

dτ Z 1

0

dσδ VⁱV^j∂iX^µ∂jX^νηµν = (6.11)

= Z τ_f

τi

dτ Z 1

0

dσδ VⁱV^j ∂_iX^µ∂_jX^νη_µν+ VⁱV^jδ (∂_iX^µ∂_jX^νη_µν) . (6.12) We will now look at the integrand terms one at the time.

δ VⁱV^j ∂_iX^µ∂_jX^νη_µν = V^j∂_iX^µ∂_jX^νη_µνδVⁱ+Vⁱ∂_iX^µ∂_jX^νη_µνδV^j = 2V^j∂_iX^µ∂_jX^νη_µνδVⁱ. (6.13) The next term will be easier to investigate under the integral sign. We restrict ourselves to variations δX^µ which vanished for τ = τ_i and τ = τ_f.

Z τf

τi

dτ Z 1

0

dσVⁱV^jδ (∂_iX^µ∂_jX^νη_µν) = (6.14)

=

Z τf

τi

dτ Z 1

0

dσVⁱV^j∂jX^ν∂iδX^µηµν + VⁱV^j∂iX^µ∂jδX^νηµν = (6.15)

= 2

Z τ_f τi

dτ Z 1

0

dσVⁱV^j∂_jX^ν∂_iδX^µη_µν = (6.16)

= 2

Z τf

τi

dτ Z 1

0

dσ∂i VⁱV^j∂_jX^νδX^µη_µν − ∂i VⁱV^j∂_jX^ν δX^µη_µν . (6.17) So, the total expression for the variation of the action is,

10The exact meaning of Dirichlet conditions will not be discussed here since they break the important Poincar´e symmetry.

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2 Z τf

τi

dτ Z 1

0

dσ∂i VⁱV^j∂jX^νδX^µηµν − ∂i VⁱV^j∂jX^ν δX^µηµν+ V^j∂iX^µ∂jX^νηµνδVⁱ . (6.18) Now we can start identifying what needs to be zero. We immediately see that we recover the already known field equations, namely,

∂_i VⁱV^j∂_jX^µ

= 0, (6.19)

V^jγ_ij = 0. (6.20)

But we are left with one more term, which we also have to force to be zero for arbitrary variations, namely,

∂_i VⁱV^j∂_jX^νδX^µη_µν = 0. (6.21) Since this is a full derivative, it clearly vanishes in the case i = 0, due to our restriction of variations. For i = 1 on the other hand, we need to impose the following (again using the fact that it is a full derivative),

V¹V^j∂_jX_µδX^µ

σ=0,1 = 0. (6.22)

But it is clear that in the gauge Vⁱ = (1, 0) (which is called the transverse gauge, a tensionless analogy to the conformal gauge) this is already imposed, since V¹ = 0. If one where to choose another gauge, for example V = (0, 1), we would have to demand [V^j∂_jX_µ]_σ=0,1 = 0. This actually gives us, in the tensile theory, the Neumann boundary conditions for the open string,

(X^µ)⁰(τ, 0) = (X^µ)⁰(τ, 1) = 0, (6.23) which physical meaning is that the endpoints of the string move at the speed of light.

Trying to interpret the physical meaning of the tensionless theory in the transversal gauge, it is of great help to compare the field equations of the tensionless string (A.78) and the equations of motion for the massless particle (6.6). By doing so, one can see that the endpoints of the string are constrained to move in a direction transversal, i.e. orthogonal, to the string and parallell to each other. Hence, it can be modelled as a collection of massless particles distributed at each position σ.

7 Outlook and Further Investigations

Considering the result obtained here, it is clear that the next step would either be a quantization or a supersymmetrization of the theory. Supersymmetrization is necessary since the theory so far have been purely bosonic, i.e. the existence of fermions have not been allowed. In other words, this kind of theory is rather unrealistic if not supersymmetrized [1].

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Globally, the case of the free string has been supersymmetrized in [2]. Also, in [12] the author discusses how this symmetrization can be described. Exactly how this works, and what he suggests, is beyond the scope of this thesis.

Quantization is a widely discussed matter and this is something that has been done in several cases, e.g. in [10], in which the author also claims that the quantization was first discussed in [13]. For a string subject to a background, the quantization is also discussed, e.g. in [3]. The quantization will not be dealt with here explicitly. Noteworthy is though, that the author of [4] is claiming that different results arise from different quantization methods, which most certainly must be investigated further.

As mentioned in the beginning, there is still ongoing research within the field of a tensionless string theory. In [5], it is discussed how Quantum Electrodynamics can be interpreted as a tensionless spinning string theory with contact interaction, i.e. what was discussed above. In fact, the desire to understand how the connection between string theory and Q.E.D. worked, was what triggered the whole project. For sure, further studies will be done and hopefully some great and interesting results will arise from it!

Acknowledgements

Firstly, I would like to thank my supervisor, Ulf Lindstr¨om, for suggesting this project and providing help and discussions when possible, even though we never seemed to be in the same country at the same time. Secondly, many thanks to my course mates, in particular Hans Nguyen and Lukas Rødland with whom I always could discuss even the weirdest questions imaginable. Lastly, I would like to thank Sergio Vargas Avila for providing useful and essential discussions, literature and feedback in all manners possible. Without him, I would not have been able to reach as far as I did.

A Detailed Calculations

Since receiving the equations of motion for all the desired cases generated a lot of long and tedious calculations, they are presented in detail here. The conventions presented in the different cases are assumed from here and onwards.

A.1 Free Relativistic Particle

The action of a free relativistic particle can be written on the following form:

S = M Z q

− ˙X²dt (A.1)

giving us the following Lagrangian:

L = M q

− ˙X² (A.2)

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Here M is the mass of the particle and ˙X^µ is the time derivative of the generalized coordinate X^µ. From (A.2) it is easy to extract the conjugated momenta P using basic knowledge of analytical mechanics, namely:

P_µ = ∂L

∂ ˙X^µ = M −2 ˙X_µ p− ˙X²

(A.3) From this expression we derive a constraint which shall be satisfied. Squaring the expression for P gives us:

P²+ M² = 0 (A.4)

Through the usual Legendre transform, one may now move to phase space using the Hamil- tonian formalism. In this particular case, we will receive a rather interesting naive Hamil- tonian, namely:

H₀ = P_µX˙^µ− L = − M ˙X² p−X˙ ²

− M q

− ˙X² = (A.5)

= M q

− ˙X²− M q

− ˙X² = 0 ⇒ (A.6)

H0 = 0 (A.7)

So we conclude that the naive Hamiltonian H₀vanishes.¹¹ Using the theory of Lagrange undetermined multiplier, we can construct another Hamiltonian, namely

H = λ P²+ M²

(A.8) Here λ is a Lagrange multiplier. Now one can formulate a phase space action, using the Hamiltonian in (A.8) and the usual Legendre transform. We get:

S^{P S} = Z

P_µX˙^µ− λ P²+ M² dt (A.9) Using Hamiltons canonical equations, another expression for the momenta is received.

X˙^µ = ∂H

∂P_µ = 2λP^µ⇒ (A.10)

P^µ = Xµ˙

2λ (A.11)

Replacing the momenta by the expression in (A.11) , i.e. integrating out the momenta, we receive the following action in configuration space:

11In fact, this is true for any so-called diffeomorphism invariant theory, meaning that the action is invariant under diffeomorphisms

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S^CS = Z



 X˙²

2λ − λ



 X˙ 2λ

!2

+ M²







dt =

Z X˙²

4λ − λM²

!

dt (A.12)

Now we have two different actions from which we can derive the equations of motions for the free relativistic particle, one set of equation describing it in phase space and the there on in configuration space. One could use the well-known Euler-Lagrange equations and Hamiltons canonical equations, but I prefer to derive those for this case using the Principle of least action, meaning that the actions should remain the same under small variations.

Hence, some variational calculus will be used.

In phase space, using (A.9), we get:

0 = δS^{P S} = δ Z

P_µX˙^µ− λ P²+ M² dt = (A.13)

= Z h

δ(P_µ) ˙X^µ+ P_µδ ˙X^µ− δλ P²+ M² − 2P^µλδP_µi

dt = (A.14)

= Z h

−δλ P²+ M² + δP_µ ˙X^µ− 2P^µλ

+ P_µδ ˙X^µ i

dt (A.15)

and we can see that we re-produce the earlier derived constraint (A.4) and the relation between the momenta and the generalized velocity (A.11) , since we need the integral to equal zero. Furthermore, we get the following relation:

Z

P_µδ ˙X^µdt = 0 (A.16)

Using the fact that δ ˙X^µ= _dt^dδX^µ, and integration by parts, we get:

0 = Z

P_µδ ˙X^µdt = Z

P_µ d

dtδX^µdt = [P_µδX^µ]^t_t^f

i −

Z

δX^µP˙_µdt (A.17) Since the first term is bound to equal zero due to the Principle of least action (for physical trajectories we set the variation for the coordinate to always equal zero at the endpoints of the trajectory). Hence, we conclude that:

P˙_µ = 0 (A.18)

P² + M² = 0 (A.19)

X˙^µ = 2P^µλ (A.20)

In configuration space, using (A.12), we get:

Limits of Relativistic Systems Bachelor Degree Project