⬅️

Physics

Introduction to Physics

Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. The most famous equation demonstrating the mass-energy equivalence is:

$$E = mc^2$$

School Physics

Subtopics can be accessed from the sidebar

Units & Measurement

The foundational study of physical quantities, the international system of units (SI), dimensional analysis, and error calculation.

Motion in a Straight Line

The kinematics of objects moving in one dimension, exploring concepts like displacement, velocity, and acceleration over time.

Motion in a Plane

The study of two-dimensional kinematics, introducing vectors, projectile motion, and uniform circular motion.

Laws of Motion

Newton's three fundamental laws governing classical mechanics, including inertia, force, and action-reaction pairs.

Work, Energy and Power

The relationship between forces applied over a distance, the conservation of mechanical energy, and the rate of doing work.

Center of Mass and Collision

Analyzing systems of particles, defining the center of mass, and understanding the conservation of momentum in elastic and inelastic collisions.

Rotational Motion

The kinematics and dynamics of rigid bodies rotating about a fixed axis, including torque and angular momentum.

Gravitation

Newton's law of universal gravitation, Kepler's laws of planetary motion, and the concept of gravitational potential energy.

Properties of Matter

The mechanical properties of solids and fluids, including elasticity, surface tension, viscosity, and Bernoulli's principle.

Heat and Thermodynamics

The study of thermal expansion, calorimetry, heat transfer, and the fundamental laws of thermodynamics governing energy and entropy.

Oscillations

The physics of periodic motion, focusing heavily on Simple Harmonic Motion (SHM) in springs and pendulums.

Waves

The propagation of mechanical disturbances through mediums, exploring concepts like wavelength, frequency, superposition, and the Doppler effect.

Electrostatics

Electrostatics deals with the study of forces, fields, and potentials arising from static charges. This forms the foundation for electromagnetism.

1. Coulomb's Law

Coulomb's law states that the electrostatic force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them.

$$F = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2}$$

Vector Form: The force on charge $q_2$ due to $q_1$ is given by:

$$\vec{F}_{12} = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r^2} \hat{r}_{12}$$

Where $\epsilon_0$ is the permittivity of free space ($\approx 8.854 \times 10^{-12} \text{ C}^2\text{N}^{-1}\text{m}^{-2}$).

2. Electric Field and Potential

Electric Field (E): The electrostatic force experienced per unit positive test charge. For a point charge $q$ at a distance $r$:

$$\vec{E} = \frac{\vec{F}}{q_0} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r}$$

Electric Potential (V): The work done by an external force in bringing a unit positive charge from infinity to a point. For a point charge $q$:

$$V = \frac{1}{4\pi\epsilon_0} \frac{q}{r}$$

Relation between E and V: The electric field is the negative gradient of the electric potential.

$$\vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}\hat{i} + \frac{\partial V}{\partial y}\hat{j} + \frac{\partial V}{\partial z}\hat{k} \right)$$

3. Electric Dipole

A system of two equal and opposite charges ($+q$ and $-q$) separated by a small distance ($2a$). The dipole moment $\vec{p}$ is a vector directed from $-q$ to $+q$ with magnitude $p = q(2a)$.

Torque in a Uniform Electric Field: When placed in a uniform field $\vec{E}$, the net force is zero, but the charges experience equal and opposite forces creating a couple. The torque ($\tau$) is the cross product of dipole moment and electric field:

$$\vec{\tau} = \vec{p} \times \vec{E} \implies \tau = pE \sin\theta$$

Potential Energy of a Dipole: The work done in rotating a dipole from an angle $\theta_1$ to $\theta_2$ is stored as potential energy ($U$). Taking standard reference at $90^\circ$:

$$U = -\vec{p} \cdot \vec{E} = -pE \cos\theta$$

4. Gauss's Law

Electric Flux ($\Phi_E$): The measure of electric field lines crossing a given surface area. For a uniform electric field and a planar area vector $\vec{A}$:

$$\Phi_E = \int \vec{E} \cdot d\vec{A}$$

Gauss's Theorem: The total electric flux through any closed surface is equal to $1/\epsilon_0$ times the net charge enclosed by the surface.

$$\oint \vec{E} \cdot d\vec{A} = \frac{q_{\text{enclosed}}}{\epsilon_0}$$

Exam Note: Gauss's Law is primarily useful for finding electric fields for highly symmetric charge distributions (e.g., infinite line charge, infinite plane sheet, spherical shells).

Current Electricity

The study of electric current in conductors, encompassing the dynamics of moving charges, resistance, and complex circuit analysis.

1. Electric Current and Drift Speed

Electric current is the rate of flow of charge. In a conductor, electrons move with an average velocity called drift speed ($v_d$) when an electric field is applied.

The relation between current ($I$) and drift speed is:

$$I = neAv_d$$

Where $n$ is number density of electrons, $e$ is elementary charge, and $A$ is the cross-sectional area.

2. Ohm's Law and Resistivity

Ohm's Law states that voltage across a conductor is directly proportional to the current flowing through it, provided temperature remains constant.

$$V = IR$$

The microscopic form relating current density ($\vec{J}$) and electric field ($\vec{E}$) is $\vec{J} = \sigma\vec{E}$, where $\sigma$ is conductivity.

Temperature Dependence of Resistivity: Resistivity ($\rho$) changes with temperature ($T$) according to the empirical formula:

$$\rho = \rho_0 [1 + \alpha(T - T_0)]$$

Where $\alpha$ is the temperature coefficient of resistivity.

3. Kirchhoff's Laws

Essential for analyzing complex electrical circuits:

  • Junction Rule (KCL): The algebraic sum of currents at any junction is zero ($\sum I = 0$). This is based on the conservation of charge.
  • Loop Rule (KVL): The algebraic sum of changes in potential around any closed loop is zero ($\sum \Delta V = 0$). This is based on the conservation of energy.

4. Wheatstone Bridge

A specific arrangement of four resistors used to determine an unknown resistance. In the balanced condition (no current through the galvanometer), the ratio of the arms is equal:

$$\frac{R_1}{R_2} = \frac{R_3}{R_4}$$

Capacitors

Devices designed to store electrical energy and charge in an electric field.

1. Capacitor and Capacitance

Capacitance ($C$) is the ratio of the charge ($Q$) on either conductor to the potential difference ($V$) between them.

$$C = \frac{Q}{V}$$

2. Parallel-Plate Capacitor

The most common type of capacitor. For plates of area $A$ separated by a distance $d$ in a vacuum, the calculation of capacitance yields:

$$C_0 = \frac{\epsilon_0 A}{d}$$

With a Dielectric: When a dielectric material of constant $K$ fills the space, the capacitance increases:

$$C = K C_0 = \frac{K \epsilon_0 A}{d}$$

3. Combination of Capacitors

Capacitors can be grouped in circuits to achieve a desired equivalent capacitance:

  • Series Combination: Charge remains constant across all capacitors. The equivalent capacitance is given by:

    $$\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n}$$

  • Parallel Combination: Voltage remains constant across all capacitors. The equivalent capacitance is given by:

    $$C_{eq} = C_1 + C_2 + \dots + C_n$$

4. Energy Stored and Energy Density

The work done in charging a capacitor is stored as electrostatic potential energy ($U$). It can be calculated as:

$$U = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV$$

Energy Density ($u$): The energy stored per unit volume in the electric field between the plates is:

$$u = \frac{1}{2}\epsilon_0 E^2$$

Moving Charges and Magnetism

This section explores the Definition of Magnetic Field $\vec{B}$, the Relation between Electric and Magnetic Fields, and the Motion of a Charged Particle in a Uniform Magnetic Field.

1. Magnetic Force

  • On a moving charge: A charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences a force given by $\vec{F} = q(\vec{v} \times \vec{B})$.
  • On a current-carrying wire: The Magnetic Force on a Current-carrying Wire of length $\vec{l}$ and current $I$ is calculated as $\vec{F} = I(\vec{l} \times \vec{B})$.
  • Torque: The Torque on a Current Loop is given by $\vec{\tau} = \vec{m} \times \vec{B}$, where $\vec{m}$ is the magnetic moment.

2. Magnetic Field due to a Current

  • Biot-Savart Law: Determines the magnetic field $d\vec{B}$ generated by a differential current element $Id\vec{l}$.

    $$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$$

  • Ampere's Law: States that the line integral of the magnetic field around any closed loop is equal to $\mu_0$ times the enclosed current.

    $$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{enc}}$$

3. Key Applications

  • Derivations for the Magnetic Field due to Current in a Straight Wire and the Field due to a Circular Current.
  • Calculating the Force between Parallel Currents, which is historically used to define the Ampere.
  • Determining the magnetic field inside a Solenoid and a Toroid using Ampere's Law.

Magnetism and Matter

This chapter studies Permanent Magnets, including Magnetic Poles and Bar Magnets, and the underlying Magnetic Properties of Matter.

1. Permanent Magnets & Earth's Field

  • Bar Magnets: Core concepts include calculating the Torque on a Bar Magnet Placed in a Magnetic Field, the Magnetic Field due to a Bar Magnet, and understanding Magnetic Scalar Potential.
  • Earth's Magnetism: Encompasses Terrestrial Magnetism, the Determination of Dip at a Place, and identifying a Neutral Point.
  • Gauss's Law for Magnetism: States that the net magnetic flux through any closed surface is zero ($\oint \vec{B} \cdot d\vec{A} = 0$), implying magnetic monopoles do not exist.

2. Measuring Instruments

  • Operational principles of the Tangent Galvanometer, Moving-coil Galvanometer, and Deflection Magnetometer.
  • Galvanometer modifications involve calculating and applying a Shunt resistor.

3. Magnetic Properties of Matter

  • Key Metrics: Defining Magnetization of Materials: Intensity of Magnetization, Magnetic Intensity $H$, Magnetic Susceptibility, and Permeability.
  • Classification: Studying the Properties of Dia-, Para- and Ferromagnetic Substances.
  • Laws and Effects: Exploring Curie's Law, Hysteresis, and the practical differences between Soft Iron and Steel.

Electromagnetic Induction

The study of how changing magnetic environments can induce electromotive forces and currents.

1. Fundamental Laws

  • Faraday's Law of Electromagnetic Induction: The induced emf is equal to the negative rate of change of magnetic flux.

    $$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

  • Lenz's Law: Dictates that the direction of the induced current opposes the change in magnetic flux that produced it, aligning with the conservation of energy.
  • Origin: Explaining The Origin of Induced emf and the formation of Eddy Current in bulk conductors.

2. Induction and Circuits

  • Types of Induction: Definitions and formulas for Self-induction and Mutual Induction.
  • Energy: The Energy Stored in an Inductor is given by the formula $U = \frac{1}{2}LI^2$.
  • Transient Responses: Formulating the Growth and Decay of Current in an $LR$ Circuit.
  • Applications: Understanding the mechanics of the Induction Coil.

Alternating Current

Electrical circuits where the current periodically reverses direction, analyzing impedance, resonance, and transformers.

Electromagnetic Waves

The unified propagation of electric and magnetic fields through space, and the complete electromagnetic spectrum.

Geometrical Optics

The behavior of light utilizing ray approximations to study reflection, refraction, lenses, mirrors, and optical instruments.

Wave Optics

The phenomena of light that require a wave model to explain, including interference (Young's double-slit), diffraction, and polarization.

Atoms and Nuclei

Rutherford and Bohr models of the atom, atomic spectra, and the composition, stability, and radioactive decay of atomic nuclei.

Dual Nature of Radiation and Matter

The quantum mechanical reality that light exhibits particle properties (photons) and matter exhibits wave properties (de Broglie waves).

Semiconductor Electronics

The physics of materials with conductivity between insulators and conductors, forming the basis of diodes, transistors, and modern computing.

Dynamics

To simplify the study of dynamical systems we will divide it into six main parts depending on whether we are studying single/many body, at low/high speeds and if we are in the quantum/classical limits. For three out of four forces we have experimentally verified Quantum Theory. For the fourth force of Gravity we have a very beautiful classical theory by Einstein whose Quantum Part is yet to be discovered experimentally but still we have really a very strong candidate called String Theory which is mathematically consistent and brings all forces under one formalism. For now we will be exploring these experimentally verified regimes:

  1. Single Body Non-Relativistic Classical Physics (CM)
  2. Many Body Non-Relativistic Classical Physics (SM)
  3. Single Body Non-Relativistic Quantum Physics (QM)
  4. Many Body Non-Relativistic Quantum Physics (CMP)
  5. Single Body Relativistic Classical Physics (STR+GR)
  6. Many-Body Relativistic Quantum Physics (QFT)

In any kind of Dynamics, it's helpful to define the SYSTEM, the STATE and EVOLUTION. In Newtonian Mechanics, we defined the SYSTEM as the particle acted on by a Force whose STATE (Initial) was given by pair of position and velocity vectors and finally using Newtons Laws F=ma (solving this differential equation), we can get the position and velocity at later times (final state). We can extend this idea to our six divisions.

System Framework 1. CM 2. SM 3. QM 4. CMP 5. STR+GR 6. QFT
The System A low-speed particle A probability distribution (ensemble) A wavefunction An interacting many-body lattice A massive body in curved spacetime A quantized field
State Variables (State) $$(\mathbf{x}, \mathbf{p})$$ $$\rho(\mathbf{q}, \mathbf{p})$$ $$|\psi\rangle$$ $$|\Psi\rangle$$ $$g_{\mu\nu}, x^\mu(\tau)$$ $$\hat{\phi}(x), |0\rangle$$
Dynamical Equation (Evolution) $$m\ddot{\mathbf{x}} = -\nabla V$$ $$\frac{\partial \rho}{\partial t} = \{H, \rho\}$$ $$i\hbar\frac{\partial}{\partial t}|\psi\rangle = \hat{H}|\psi\rangle$$ $$\hat{H} = -t \sum_{\langle i,j \rangle, \sigma} (\hat{c}_{i\sigma}^\dagger \hat{c}_{j\sigma} + h.c.) + U \sum_i \hat{n}_{i\uparrow} \hat{n}_{i\downarrow}$$ $$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}$$ $$\mathcal{L} = \bar{\psi}(i\gamma^\mu\partial_\mu - m)\psi$$

Single Body Non-Relativistic Classical Physics (CM)

This section explores classical mechanics frameworks (Newtonian, Lagrangian, and Hamiltonian) for macroscopic objects moving at speeds much less than the speed of light.

Many Body Non-Relativistic Classical Physics (SM)

This section explores Statistical Mechanics and Thermodynamics, analyzing the macroscopic properties of systems containing a massive number of classical particles using probability ensembles.

Single Body Non-Relativistic Quantum Physics (QM)

Here we study the wave-like behavior of single subatomic particles using the Schrödinger equation, quantum operators, and wavefunctions.

Many Body Non-Relativistic Quantum Physics (CMP)

Also known as Condensed Matter Physics, this field studies the complex emergent behaviors (like superconductivity, magnetism, and Mott insulators) that arise when many quantum particles interact in a lattice.

Single Body Relativistic Classical Physics (STR+GR)

This covers Einstein's Special and General Relativity, describing the dynamics of massive bodies at high speeds and the curvature of spacetime caused by gravity.

Many-Body Relativistic Quantum Physics (QFT)

QFT is an attempt to unify the rules of the fast world with the rules of the small world. One of the first steps in these directions can be just to pick up the classical rules of energy conservation, apply the operator formalism and see if we can keep the conservation of probability intact while maintaining the Lorentz covariance.

Deep Dive: Where do the operators come from?

Fundamentally, energy and momentum are the conserved quantities associated with time and space translation symmetries (Noether's Theorem). In quantum mechanics, the generator of time translation is the Hamiltonian, giving us the energy operator $E \to i\hbar \frac{\partial}{\partial t}$. The generator of space translation gives us the momentum operator $\mathbf{p} \to -i\hbar \nabla$. We plug these directly into classical energy equations to find our quantum equations of motion.

1. The Non-Relativistic Case (Schrödinger Equation)

We start with the classical energy conservation for a free particle:

$$ E = \frac{p^2}{2m} $$

Applying the operator formalism to a state $|\psi\rangle$ (represented by a complex wavefunction $\psi$), we get the standard Schrödinger equation:

$$ i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi $$

Doing the same for the "bra" vector (taking the complex conjugate of the equation) gives:

$$ -i\hbar \frac{\partial \psi^*}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi^* $$

To verify if probability is conserved, we calculate the time evolution of the probability density (the mod square, $\rho = \psi^*\psi$):

$$ \frac{\partial \rho}{\partial t} = \frac{\partial}{\partial t}(\psi^* \psi) = \psi^* \frac{\partial \psi}{\partial t} + \frac{\partial \psi^*}{\partial t} \psi $$

Substituting the time derivatives from our two Schrödinger equations:

$$ \frac{\partial \rho}{\partial t} = \psi^* \left( \frac{i\hbar}{2m} \nabla^2 \psi \right) - \left( \frac{i\hbar}{2m} \nabla^2 \psi^* \right) \psi $$

$$ \frac{\partial \rho}{\partial t} = \frac{i\hbar}{2m} \nabla \cdot (\psi^* \nabla \psi - \psi \nabla \psi^*) $$

By defining the probability current $\mathbf{j} = \frac{-i\hbar}{2m} (\psi^* \nabla \psi - \psi \nabla \psi^*)$, we arrive at the continuity equation:

$$ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0 $$

Because $\rho = |\psi|^2 \geq 0$, probability is strictly positive and perfectly conserved. Non-relativistic QM works beautifully!

2. The Relativistic Case (Klein-Gordon Equation)

Now, let's attempt the exact same procedure using the relativistic energy-momentum relation:

$$ E^2 = p^2c^2 + m^2c^4 $$

Applying the energy and momentum operators to a wavefunction $\phi$, we obtain the Klein-Gordon equation:

$$ -\hbar^2 \frac{\partial^2 \phi}{\partial t^2} = -\hbar^2 c^2 \nabla^2 \phi + m^2 c^4 \phi $$

Taking the complex conjugate for the bra vector:

$$ -\hbar^2 \frac{\partial^2 \phi^*}{\partial t^2} = -\hbar^2 c^2 \nabla^2 \phi^* + m^2 c^4 \phi^* $$

To check for probability conservation, we multiply the first equation by $\phi^*$, the second by $\phi$, and subtract them (effectively taking the dot product to find the conserved current):

$$ -\hbar^2 \left( \phi^* \frac{\partial^2 \phi}{\partial t^2} - \phi \frac{\partial^2 \phi^*}{\partial t^2} \right) = -\hbar^2 c^2 \left( \phi^* \nabla^2 \phi - \phi \nabla^2 \phi^* \right) $$

This can be mathematically massaged into a continuity equation $\frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{j} = 0$, but this time, the "probability density" $\rho$ evaluates to:

$$ \rho = \frac{i\hbar}{2mc^2} \left( \phi^* \frac{\partial \phi}{\partial t} - \phi \frac{\partial \phi^*}{\partial t} \right) $$

The Breakdown of Single-Particle Probability

Here lies the catastrophic failure of single-particle relativistic quantum mechanics. Because the Klein-Gordon equation is second-order in time, both the initial wavefunction $\phi$ and its time derivative $\partial_t \phi$ can be chosen completely independently.

As a direct result, the density $\rho$ can be negative!

A negative probability makes absolutely no physical sense. To resolve this paradox, we can no longer treat $\phi$ as a simple probability wavefunction. Instead, $\rho$ must be reinterpreted as a charge density (which is allowed to be positive or negative), and $\phi$ must be elevated from a state vector to a Quantum Field Operator capable of creating and annihilating particles. This mathematical necessity is the birthplace of Quantum Field Theory.

3. Dirac's Brilliant Fix (The Dirac Equation)

In 1928, Paul Dirac sought to cure the negative probability disease of the Klein-Gordon equation. He realized that to guarantee a positive probability density (like the Schrödinger equation), the new equation had to be first-order in time. However, to satisfy special relativity, space and time must be treated on equal footing, meaning the equation also had to be first-order in space.

Dirac effectively tried to take the "square root" of the relativistic energy equation $E^2 = p^2c^2 + m^2c^4$. He proposed a linear equation of the form:

$$ E = c(\alpha_1 p_1 + \alpha_2 p_2 + \alpha_3 p_3) + \beta mc^2 $$

The Algebraic Deduction: Why 4x4?

Dirac knew that if he squared his new linear equation, it had to perfectly reproduce the standard relativistic energy equation. Let's expand $E^2$:

$$ E^2 = c^2 \sum_{i=1}^3 \alpha_i^2 p_i^2 + c^2 \sum_{i < j} (\alpha_i \alpha_j + \alpha_j \alpha_i)p_i p_j + mc^3 \sum_{i=1}^3 (\alpha_i \beta + \beta \alpha_i)p_i + \beta^2 m^2c^4 $$

For this massive expansion to collapse neatly back into $E^2 = p^2c^2 + m^2c^4$, the coefficients $\alpha_i$ and $\beta$ must satisfy three strict mathematical rules:

  1. They must square to 1: $\alpha_1^2 = \alpha_2^2 = \alpha_3^2 = \beta^2 = I$
  2. They must mutually anti-commute: $\alpha_i \alpha_j + \alpha_j \alpha_i = 0$ (for $i \neq j$)
  3. $\beta$ must anti-commute with all $\alpha$'s: $\alpha_i \beta + \beta \alpha_i = 0$

Clearly, ordinary numbers cannot anti-commute ($2 \times 3$ is always $3 \times 2$). Therefore, $\alpha_i$ and $\beta$ must be matrices. But what size?

  • Because they square to $I$, their eigenvalues can only be $+1$ or $-1$.
  • Because they anti-commute, one can prove mathematically that their trace (the sum of their diagonal elements) must be strictly zero.
  • For the trace to be zero, there must be an equal number of $+1$ and $-1$ eigenvalues. This means the matrices must have an even dimension ($2 \times 2$, $4 \times 4$, $6 \times 6$, etc.).

Could they be $2 \times 2$? The physics world already knew of three mutually anti-commuting $2 \times 2$ matrices: the Pauli spin matrices ($\sigma_x, \sigma_y, \sigma_z$). However, Dirac needed four mutually anti-commuting matrices (three $\alpha$'s for spatial dimensions, and one $\beta$ for mass). Since there are only three such $2 \times 2$ matrices in existence, $2 \times 2$ is too small!

Dirac was therefore forced to move to the next smallest even dimension: $4 \times 4$ matrices.

Applying the quantum operator formalism ($E \to i\hbar \partial_t$ and $\mathbf{p} \to -i\hbar \nabla$) to Dirac's linear Hamiltonian yields the original Dirac Equation:

$$ i\hbar \frac{\partial \psi}{\partial t} = \left( -i\hbar c \boldsymbol{\alpha} \cdot \nabla + \beta mc^2 \right) \psi $$

Because the operators are $4 \times 4$ matrices, the wavefunction $\psi$ can no longer be a simple scalar function. It is forced by the math to become a 4-component column vector called a Dirac Spinor. This mathematical inevitability accidentally predicted the existence of intrinsic spin-1/2 (up and down states) and the existence of antimatter (the positron)!

The Covariant Form

To make the Lorentz symmetry obvious, we multiply the equation by $\beta / c$ and define new matrices called the Gamma matrices: $\gamma^0 = \beta$ and $\gamma^i = \beta \alpha_i$. In natural units ($\hbar = c = 1$), the Dirac equation takes its most famous, compact covariant form:

$$ (i\gamma^\mu \partial_\mu - m)\psi = 0 $$

4. Popular Bases for the Gamma Matrices

The algebra of the Gamma matrices ($\{\gamma^\mu, \gamma^\nu\} = 2\eta^{\mu\nu}$) defines the physics, but the actual $4 \times 4$ matrices can be written in different ways (bases) depending on the physical regime being studied. Here are the three most common representations:

  • The Dirac (Standard) Basis:
    Best suited for non-relativistic limits. In this basis, $\gamma^0$ is diagonal. It perfectly separates the spinor into a "large" upper component (representing the particle) and a "small" lower component (representing the antiparticle) at low energies.

    $$ \gamma^0 = \begin{pmatrix} I & 0 \\ 0 & -I \end{pmatrix}, \quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix} $$

  • The Weyl (Chiral) Basis:
    Best suited for ultra-relativistic limits or massless particles (like the historical view of neutrinos). Here, $\gamma^5$ is diagonal. It splits the Dirac spinor into left-handed and right-handed chiral Weyl spinors, which is essential for studying the Weak Interaction and parity violation.

    $$ \gamma^0 = \begin{pmatrix} 0 & I \\ I & 0 \end{pmatrix}, \quad \gamma^i = \begin{pmatrix} 0 & \sigma^i \\ -\sigma^i & 0 \end{pmatrix} $$

  • The Majorana Basis:
    Designed so that all Gamma matrices are purely imaginary. As a result, the Dirac equation becomes a purely real differential equation. This is used to describe Majorana fermions—particles that are their own antiparticles (a heavy area of research in modern condensed matter and neutrino physics).

$$...$$

Research Physics

Welcome to the research frontier. This section is dedicated to advanced topics, ongoing personal projects, and deep dives into the current boundaries of modern physics.

Select a topic from the sidebar to view the detailed research notes and simulations.

Simulating Hubbard Model using Exact Diagonalization and Quantum Computation

Second Quantization

In formulating quantum mechanics from classical mechanics, we elevate the dynamical variables $q_i$'s and $p_i$'s to operators and define a commutation relation between them as $[q_i,p_i]=i\hbar$. This is called First Quantization.

When we elevate the wavefunction $\psi(r)$ itself to operator also known as Field Operator $\vu{\psi}(r)$, this formulation is called Second Quantization. Similar to how a $q_i$ has a canonical conjugate as $p_i$, the field operator $\vu{\psi}(r)$ has $\vu{\psi}^{\dagger}(r)$ and they have a commutation relation between them given as $[\vu{\psi}(r_1),\vu{\psi}^{\dagger}(r_2)]=\delta(r_1 - r_2)$ for Bosons and Anti-commutation relation for Fermions i.e. $\{\vu{\psi}(r_1),\vu{\psi^{\dagger}}(r_2)\}=\delta(r_1 - r_2)$.

This field operator which is function of space and time in one dimension in general is defined as :

$$ \vu{\psi}(r)=\sum_n c_n \phi_n(r) $$

where, $\phi_n(x)$ is the usual first quantization wavefunction and $c_n$ is the annihilation operator. Similarly its canonical conjugate is defined as :

$$ \vu{\psi}^{\dagger}(r)=\sum_n c_n^{\dagger} \phi_n^*(r) $$

where, $\phi_n^*(x)$ is the complex conjugate of the wave function and $c_n^{\dagger}$ is the creation operator.

These creation and annhilation operators will act on the STATE of the system which is called the "Fock Space". which we will denote by :

$$ \ket{n_1, n_2, n_3, ..., n_i, ... , n_N} $$

where, $n_i$ tells us the number of particles present at the ith state (first quantization wavefunction).

A "Vaccum State", where physically there are no particles present in our space is $\ket{0, 0, ..., 0}$. So as an example if we act $c_1^{\dagger}$ on the vaccum state, we get $\sqrt{2} \ket{1, 0, ..., 0}$ and $c_1$ acting on vacuum state will give us zero.

Disclaimer

We have borrowed the action of ladder operators from first quantization from the harmonic oscillator but physically these are two different operators. In first quantization the creation ladder operator will change the $\ket{n}$ to $\ket{n+1}$ by and factor of $\sqrt{n+1}$ and annihilation operator will change $\ket{n}$ to $\ket{n-1}$ by a factor of $\sqrt{n}$.

System

If we look at atoms whose outermost shell is the s orbital then their wave function is not localized i.e. of the form $\psi = e^{i(k\cdot r)}$ (The probability of finding the particle is equally like everywhere). But if we take a look at the atoms whose outermost electrons lie in the d orbital. These wavefunctions are bound to the atom and they have localization around the ion. But the localization is not strong enough for them so as to stop them from tunneling the neighbouring atoms and occupy and empty state present. Now given the fixed number of such electrons and fixed number of sites, the system will choose a state such that the total energy of the system is minimized. The dynamics of such materials can be modelled by what is called the Hubbard Model whose Hamiltonian can be written as:

$$ H = -t \sum_{i,\sigma} c_{i,\sigma}^\dagger c_{i+1,\sigma} + h.c + U \sum_{i}^n {n_{i\uparrow}} {n_{i\downarrow}} $$

where:

  • $c_{i,\sigma}^\dagger$: The creation operator for an electron with spin $\sigma$ ($\uparrow$ or $\downarrow$) at site $i$.
  • $c_{i,\sigma}$: The annihilation operator for an electron with spin $\sigma$ at site $i$.
  • $t$: The hopping amplitude, representing the kinetic energy of an electron hopping between nearest-neighbor sites.
  • $n_{i\sigma} = c_{i,\sigma}^\dagger c_{i,\sigma}$: The number operator for electrons with spin $\sigma$ at site $i$.
  • $U$: The on-site Coulomb interaction energy between two electrons of opposite spins ($\uparrow$ and $\downarrow$) at the same site.

It is an extension of the Tight Binding Approximation for single band and inclusion of electron-electron repulsion. Hubbard Model was introduced in order to explain the insulating behaviour of partially filled Transition Metal Oxide like TiO. Not only that in the limiting behaviour where the repulsion between electrons is more than the tunneling term, the electrons tends to localize on sites having same spin and hence show ferromagnetism.

State

We are going to choose our basis state as what is called Wernier Wave Functions (they are fourier transforms of conventional Bloch States) which will give us the state of ONE electron localized on ONE particular site labelled by index i.

$$ W_\gamma(\vec{r}-\vec{R_i}) = \frac{1}{\sqrt{N}} \sum_{k} e^{-i \vec{k}\cdot\vec{R_i}} \phi_{k,\gamma}(\vec{r}) $$

Here, $\gamma$ is the band index, $\vec{R}$ is the ionic position vector, $\phi(\vec{r})$ is the Bloch wave function. We can then also define a ortho normality between states of different sites.

$$ \bra{W_n(\vec{r}-\vec{R_i})}\ket{W_n(\vec{r}-\vec{R_j})} = \delta_{R_i,R_j} $$

Since this is a fermionic system, each site can have either one up spin or one down spin or both one up and one down spin. In order to denote the state of the system i.e. all sites at once we will use the notation as follows:

$$ \ket{\uparrow} \quad \ket{\downarrow} \quad \ket{\uparrow\downarrow} $$

Exact Diagonalization

2 Sites 2 Electrons

All possible states for 2 sites and 2 electrons will look like:

$$ \ket{\uparrow,\uparrow} \quad \ket{0,\uparrow\downarrow} \quad \ket{\uparrow, \downarrow} \quad \ket{\downarrow,\uparrow} \quad \ket{\uparrow\downarrow, 0} \quad \ket{\downarrow,\downarrow} $$

The Hamiltonian for this system is:

$$ \begin{aligned} H = &-t ( c_{1\uparrow}^\dagger c_{2\uparrow} + c_{2\uparrow}^\dagger c_{1\uparrow} + c_{1\downarrow}^\dagger c_{2\downarrow} + c_{2\downarrow}^\dagger c_{1\downarrow} ) \\ &+ U ( n_{1\uparrow} n_{1\downarrow} + n_{2\uparrow} n_{2\downarrow} ) \end{aligned} $$

When we operate this Hamiltonian operator on all the six states we get:

$H \ket{\uparrow,\uparrow} = 0$

$H \ket{0,\uparrow\downarrow} = -t ( \ket{\uparrow, \downarrow} + \ket{\downarrow,\uparrow} ) + U (\ket{0,\uparrow\downarrow})$

$H \ket{\downarrow, \uparrow} = -t ( \ket{0,\uparrow\downarrow} + \ket{\uparrow\downarrow, 0} )$

$H \ket{\downarrow\uparrow, 0} = -t ( \ket{\downarrow,\uparrow} + \ket{\uparrow, \downarrow}) + U \ket{\uparrow\downarrow, 0}$

$H \ket{\downarrow,\downarrow} = 0$

From this we can easily get the matrix element. For t = 1, U = 1:

$$ \begin{bmatrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & -1 & -1 & 0 & 0 \\ 0 & -1 & 0 & 0 & -1 & 0 \\ 0 & -1 & 0 & 0 & -1 & 0 \\ 0 & 0 & -1 & -1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} $$

On Diagonalizing this matrix numerically using LAPACK library function, we get Eigenvalues : 2.56, -1.56, 1, 0, 0, 0. The Ground State Energy as -1.56.

Nagaoka's Theorem

Statement: In Hubbard Model, the relative ground state in the sector with one unoccupied site is maximally FERROMAGNETIC for large U.

Nagaoka's Theorem Graph

For 4 sites and 3 electrons we have plotted U v/s Ground State Energies

In operational sense it means that if we take one less than half filled sites, then for large value of U (interaction between electrons on same site), the ground state of the system will be Ferromagnetic and spins will try to allow themselves in one direction. The Hubbard model can become equivalent to the Heisenberg model in the limit of strong electron-electron interactions (U) compared to the kinetic energy (t), particularly in the limit of large U and at half-filling of the electron density (one electron per lattice site). In this regime, virtual hopping processes are suppressed due to the high energy cost of double occupancy, leading to an effective spin-spin interaction between localized electrons. Under these conditions, the Hubbard model effectively reduces to the Heisenberg model, where the kinetic energy terms are neglected, and the dominant interactions are described by spin operators.

Quantum Simulation

Qubit

In quantum computing, the fundamental unit of information is the qubit, which can exist in a superposition of two basis states $|0\rangle$ and $|1\rangle$. The state of a single qubit is generally represented as:

$$ |\psi\rangle = \alpha |0\rangle + \beta |1\rangle $$

where:

  • $|0\rangle$ and $|1\rangle$ are the computational basis states, analogous to the classical binary states.
  • $\alpha$ and $\beta$ are complex numbers representing the probability amplitudes of the respective basis states.
  • The coefficients satisfy the normalization condition $|\alpha|^2 + |\beta|^2 = 1$, ensuring the total probability is conserved.
Bloch Sphere

Bloch Sphere

A general qubit in this 3 dimensional space can then be written as:

$$ |\psi\rangle = \cos\left(\frac{\theta}{2}\right) |0\rangle + e^{i\phi} \sin\left(\frac{\theta}{2}\right) |1\rangle $$

where:

  • $\theta$ is the polar angle ($0 \leq \theta \leq \pi$).
  • $\phi$ is the azimuthal angle ($0 \leq \phi < 2\pi$).
  • The coefficients $\alpha$ and $\beta$ are parameterized as: $\alpha = \cos\left(\frac{\theta}{2}\right)$ and $\beta = e^{i\phi} \sin\left(\frac{\theta}{2}\right)$.

In the context of quantum computing, the $|0\rangle$ and $|1\rangle$ states of a qubit can be mapped to the physical occupation states of a fermionic site:

  • $|0\rangle$: Represents the absence of a particle (no occupation) at the site.
  • $|1\rangle$: Represents the presence of a particle (occupied) at the site.

For the Hubbard Model, which includes both spin-up ($\uparrow$) and spin-down ($\downarrow$) fermions at each site, the Hilbert space of a single site is expanded to account for four possible states:

  • $|00\rangle$: No particles at the site.
  • $|01\rangle$: A spin-down ($\downarrow$) particle occupies the site.
  • $|10\rangle$: A spin-up ($\uparrow$) particle occupies the site.
  • $|11\rangle$: Both spin-up ($\uparrow$) and spin-down ($\downarrow$) particles occupy the site.

Jordan Wigner Transformation

The Jordan-Wigner Transformation allows fermionic creation and annihilation operators to be represented as spin operators. Specifically, the creation and annihilation operators at site $j$ are expressed as:

$$ \begin{aligned} c_{j}^\dagger &= \frac{1}{2} \left( \prod_{k=1}^{j-1} Z_k \right) (X_j - i Y_j) \\ c_{j} &= \frac{1}{2} \left( \prod_{k=1}^{j-1} Z_k \right) (X_j + i Y_j) \end{aligned} $$

Variational Quantum Eigensolver

The ground state energy of a quantum system can be expressed as the expectation value of the Hamiltonian $H$ with respect to a trial wavefunction $|\psi(\theta)\rangle$, parameterized by $\theta$. Mathematically, this is given by:

$$ E(\theta) = \langle \psi(\theta) | H | \psi(\theta) \rangle $$

where $E(\theta)$ represents the energy for a given parameter set $\theta$. The Variational Quantum Eigensolver aims to minimize this energy by optimizing the parameters $\theta$ such that:

$$ E_{\text{ground}} = \min_{\theta} \langle \psi(\theta) | H | \psi(\theta) \rangle $$

Variational Quantum Eigensolver

Variational Quantum Algorithm Eigensolver

The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm designed to find the ground state energy of quantum systems. The algorithm minimizes the expectation value of the Hamiltonian $H$, which represents the system's energy operator, to identify the lowest possible energy state.

Superconductivity

Superconductivity is a phenomenon of exactly zero electrical resistance and expulsion of magnetic flux fields occurring in certain materials when cooled below a characteristic critical temperature.

BCS Theory & Cooper Pairs

Currently developing notes on the microscopic theory of superconductivity, focusing on electron-phonon interactions and the formation of Cooper pairs.

String Theory

A theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

Fundamental Strings & Extra Dimensions

Currently developing notes on the quantization of the relativistic string, D-branes, and the necessity of extra spatial dimensions for mathematical consistency.

Physics Exercises Archive

Welcome to the testing frontier. Select a topic from the sidebar to practice chapter-wise Previous Year Questions (PYQs) for TIFR, JEST, and NET.

Mathematical Physics - TIFR PYQs

Q1 (2013). The product $AB$ of two Hermitian matrices $A$ and $B$ is anti-Hermitian[cite: 7]. It follows that[cite: 7]:
Correct Answer: A

Explanation:
By definition, a matrix is Hermitian if it is equal to its own conjugate transpose. Thus, we are given $A^\dagger = A$ and $B^\dagger = B$.

We are also given that their product $AB$ is anti-Hermitian, which means taking its conjugate transpose yields the negative of the matrix: $(AB)^\dagger = -(AB)$.

Using the matrix property that $(AB)^\dagger = B^\dagger A^\dagger$, we substitute our knowns:
$$ B^\dagger A^\dagger = -(AB) $$
$$ BA = -AB $$
$$ AB + BA = 0 $$
The expression $AB + BA$ is the definition of the anticommutator, denoted as $\{A, B\}$. Therefore, $\{A, B\} = 0$.

Classical Mechanics - TIFR PYQs

Q1 (2010). A high-velocity missile, travelling in a horizontal line with a kinetic energy of $3.0 \text{ GigaJoules (GJ)}$, explodes in flight and breaks into two pieces A and B of equal mass[cite: 3]. One of these pieces (A) flies off in a straight line perpendicular to the original direction in which the missile was moving and its kinetic energy is found to be $2.0 \text{ GJ}$[cite: 3]. If gravity can be neglected for such high-velocity projectiles, it follows that the other piece (B) flew off in a direction at an angle with the original direction of[cite: 3]:
Correct Answer: A

Explanation:
Let the initial mass of the missile be $2m$. The two pieces A and B each have mass $m$.
Using $E = \frac{p^2}{2 \times \text{mass}}$, we can relate kinetic energy to momentum:
Initial momentum (horizontal, say x-axis): $p_0 = \sqrt{2(2m)E_0} = \sqrt{4m(3)} = \sqrt{12m}$.
Momentum of piece A (perpendicular, say y-axis): $p_A = \sqrt{2m E_A} = \sqrt{2m(2)} = \sqrt{4m}$.

By conservation of linear momentum:
x-axis: $p_0 = p_{Bx} \implies p_{Bx} = \sqrt{12m}$
y-axis: $0 = p_A + p_{By} \implies p_{By} = -p_A = -\sqrt{4m}$

The angle $\theta$ that piece B makes with the original horizontal direction is given by:
$$ \tan\theta = \left| \frac{p_{By}}{p_{Bx}} \right| = \frac{\sqrt{4m}}{\sqrt{12m}} = \frac{1}{\sqrt{3}} $$
Thus, $\theta = 30^\circ$.

Quantum Mechanics - TIFR PYQs

Q1 (2011). Two identical non-interacting particles, each of mass $m$ and spin $1/2$, are placed in a one-dimensional box of length $L$[cite: 4]. In quantum mechanics, the lowest possible value of the total energy of these two particles is $\epsilon_0$[cite: 4]. If, instead, four such particles are introduced into a similar one-dimensional box of length $2L$, then the lowest possible value of their total energy will be[cite: 4]:
Correct Answer: B

Explanation:
The energy levels of a particle in a 1D box of length $L$ are $E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2}$.
Since the particles are spin-1/2 fermions (electrons), the Pauli exclusion principle dictates that at most two particles (spin up, spin down) can occupy each energy level.

For 2 particles in a box of length $L$, both can occupy the ground state ($n=1$).
$$ \epsilon_0 = 2 \times E_1 = 2 \times \frac{\pi^2 \hbar^2}{2mL^2} = \frac{\pi^2 \hbar^2}{mL^2} $$
Now, consider a box of length $2L$. The new energy levels are $E'_n = \frac{n^2 \pi^2 \hbar^2}{2m(2L)^2} = \frac{n^2}{8} \left( \frac{\pi^2 \hbar^2}{mL^2} \right) = \frac{n^2}{8}\epsilon_0$.

For 4 particles in this new box, two will occupy $n=1$, and two will occupy $n=2$.
$$ E_{\text{total}} = 2(E'_1) + 2(E'_2) = 2\left(\frac{1}{8}\epsilon_0\right) + 2\left(\frac{4}{8}\epsilon_0\right) $$
$$ E_{\text{total}} = \frac{1}{4}\epsilon_0 + \epsilon_0 = \frac{5}{4}\epsilon_0 $$
Q2 (2014). Consider the Hamiltonian $\hat{H} = f \vec{x} \cdot \vec{\sigma}$[cite: 9]. Here $\vec{x}$ is the position vector, $f$ is a constant and $\vec{\sigma} = (\sigma_x, \sigma_y, \sigma_z)$ , where $\sigma_x, \sigma_y, \sigma_z$ are the three Pauli matrices[cite: 9]. The energy eigenvalues are[cite: 9]:
Correct Answer: C

Explanation:
By expanding the dot product $\vec{x} \cdot \vec{\sigma}$, we multiply the position coordinates by their respective Pauli matrices:
$$ H = f(x\sigma_x + y\sigma_y + z\sigma_z) $$ $$ H = f \left[ x\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} + y\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix} + z\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \right] $$ $$ H = f \begin{pmatrix} z & x - iy \\ x + iy & -z \end{pmatrix} $$
To find the eigenvalues $\lambda$, we solve the characteristic equation $\det(H - \lambda I) = 0$:
$$ (fz - \lambda)(-fz - \lambda) - f^2(x - iy)(x + iy) = 0 $$ $$ -f^2z^2 + \lambda^2 - f^2(x^2 + y^2) = 0 $$ $$ \lambda^2 = f^2(x^2 + y^2 + z^2) $$
Taking the square root gives $\lambda = \pm f\sqrt{x^2 + y^2 + z^2}$.

Heat & Statistical Mechanics - TIFR PYQs

Q1 (2012). Consider a sealed but thermally conducting container of total volume $V$, which is in equilibrium with a thermal bath at temperature $T$[cite: 5]. The container is divided into two equal chambers by a thin but impermeable partition[cite: 5]. One of these chambers contains an ideal gas, while the other half is a vacuum[cite: 5]. If the partition is removed and the ideal gas is allowed to expand and fill the entire container, then the entropy per molecule of the system will increase by an amount[cite: 5]:
Correct Answer: C

Explanation:
This represents a free expansion of an ideal gas. The initial volume occupied by the gas is $V_i = V/2$, and the final volume is $V_f = V$.
The change in entropy for an ideal gas expanding isothermally into a vacuum is given by:
$$ \Delta S = n R \ln\left(\frac{V_f}{V_i}\right) = N k_B \ln\left(\frac{V}{V/2}\right) = N k_B \ln(2) $$
where $N$ is the total number of molecules and $k_B$ is the Boltzmann constant.
The question asks for the entropy increase *per molecule*, so we divide by $N$:
$$ \Delta s = \frac{\Delta S}{N} = k_B \ln 2 $$

Solid State & Condensed Matter - TIFR PYQs

Q1 (2015). In the basic band structure theory of crystalline solids, which of the following leads to energy gaps in the allowed electronic energy values?[cite: 11]
Correct Answer: B

Explanation:
In the nearly-free electron model, electrons moving through a periodic crystal lattice behave mostly as free waves. However, when the electron's wave vector $k$ approaches the Brillouin zone boundaries (i.e., $k \approx \pm \pi/a$), the Bragg reflection condition is satisfied.

The forward-moving and backward-reflected waves interfere to produce standing waves, splitting the continuous energy parabolic dispersion into two distinct states (one with probability peaks at the ion cores, and one peaking between them). The difference in potential energy between these two standing wave states creates the forbidden energy band gaps.

1. The Relativistic Point Particle

Consider a point particle propagating in a $d$-dimensional target spacetime, tracing out a 1D worldline $x^\mu(\tau)$ where $\tau$ is the proper time. In classical physics, the principle of least action dictates that a free particle follows the path of extremal length. The proper length interval is $dl = \sqrt{-ds^2} = \sqrt{-\eta_{\mu\nu}dx^\mu dx^\nu}$.

The action is strictly proportional to the length of this worldline, with the particle's mass $m$ serving as the proportionality constant:

$$ S[x] = -m \int dl = -m \int d\tau \sqrt{-\eta_{\mu\nu} \dot{x}^\mu \dot{x}^\nu} $$

Note: The negative sign inside the square root ensures a real value, as massive particles travel on timelike trajectories ($ds^2 < 0$). The negative sign outside ensures that maximizing the proper time (the straightest path in Minkowski space) minimizes the classical action.

Varying this action yields the canonical momentum $p_\mu = \frac{\partial L}{\partial \dot{x}^\mu} = \frac{m\dot{x}_\mu}{\sqrt{-\dot{x}^2}}$. Squaring this momentum vector inherently produces the classical mass-shell constraint:

$$ p^\mu p_\mu = \frac{m^2 \dot{x}^\mu \dot{x}_\mu}{-\dot{x}^\alpha \dot{x}_\alpha} = -m^2 \implies p^2 + m^2 = 0 $$

Upon applying Dirac quantization constraints ($\hat{p}_\mu \to -i\partial_\mu$), we demand that physical quantum states are annihilated by this constraint: $(\hat{p}^2 + m^2)|\psi\rangle = 0$, flawlessly recovering the Klein-Gordon equation.

2. The Bosonic String Actions

We now elevate our 0D point particle to a 1D string. As it moves through time, it sweeps out a 2D surface called a worldsheet, parameterized by $\sigma^a = (\tau, \sigma)$. The target spacetime coordinates are now scalar fields on this worldsheet: $X^\mu(\tau, \sigma)$.

The Nambu-Goto Action

Generalizing the point particle, the string's action must be proportional to the proper area of the worldsheet. Geometrically, the area of a parallelogram spanned by two vectors $\vec{a}$ and $\vec{b}$ is $A = |\vec{a}||\vec{b}|\sin\theta$. Using $\sin^2\theta = 1 - \cos^2\theta$, we can rewrite this strictly using dot products:

$$ A = \sqrt{|\vec{a}|^2|\vec{b}|^2 - (\vec{a} \cdot \vec{b})^2} $$

On the worldsheet, the tangent vectors are $\partial_\tau X^\mu$ and $\partial_\sigma X^\mu$. Their dot products form the induced metric $h_{ab} = \eta_{\mu\nu} \partial_a X^\mu \partial_b X^\nu$. The term under the square root is exactly the negative determinant of this metric. Therefore, the Nambu-Goto action is:

$$ S_{NG} = -T \int d\tau d\sigma \sqrt{-\det(h_{ab})} $$

Here, $T = \frac{1}{2\pi\alpha'}$ is the string tension, where $\alpha'$ is the Regge slope.

The Polyakov Action

The Nambu-Goto action is highly non-linear due to the square root, making path-integral quantization notoriously difficult. To solve this, Polyakov introduced an independent, auxiliary worldsheet metric $\gamma_{ab}(\tau, \sigma)$ to eliminate the square root:

$$ S_{P} = -\frac{T}{2} \int d^2\sigma \sqrt{-\gamma} \, \gamma^{ab} \partial_a X^\mu \partial_b X_\mu $$

3. Symmetries and the Conformal Gauge

The Polyakov action is invariant under three critical symmetries:

  1. Global Poincaré Invariance: Spacetime translations and rotations ($X^\mu \to \Lambda^\mu_\nu X^\nu + a^\mu$).
  2. Local Diffeomorphism: Worldsheet coordinate reparameterizations.
  3. Weyl Invariance: Local scale transformations of the metric: $\gamma_{ab} \to e^{2\omega(\tau, \sigma)}\gamma_{ab}$.

The Magic of 2D Weyl Invariance

Why does Weyl scaling work so perfectly? If $\gamma_{ab} \to e^{2\omega}\gamma_{ab}$, the inverse metric scales as $\gamma^{ab} \to e^{-2\omega}\gamma^{ab}$. Meanwhile, the determinant $\gamma$ scales as $e^{4\omega}\gamma$, so its square root scales as $\sqrt{-\gamma} \to e^{2\omega}\sqrt{-\gamma}$. Plugging this into the action: $(e^{2\omega})(e^{-2\omega}) = 1$. The scaling factors exactly cancel out! This remarkable cancellation is unique to 2-dimensional worldsheets.

A $2 \times 2$ symmetric metric $\gamma_{ab}$ has 3 independent components. We can use our 2 diffeomorphism freedoms and 1 Weyl freedom to completely fix the metric to the flat Minkowski metric multiplied by a conformal factor: $\gamma_{ab} = e^\phi \eta_{ab}$. Because of Weyl invariance, $e^\phi$ drops out entirely, leaving the Conformal Gauge:

$$ S = \frac{T}{2} \int d\tau d\sigma (\dot{X}^2 - X'^2) $$

The equations of motion are simply the 2D wave equation: $(\partial_\tau^2 - \partial_\sigma^2)X^\mu = 0$.

However, fixing the gauge means we must impose the equations of motion for the auxiliary metric $\gamma_{ab}$ as manual constraints. Varying the action with respect to $\gamma^{ab}$ yields the worldsheet stress-energy tensor $T_{ab} = 0$, giving us the Virasoro Constraints:

$$ \dot{X} \cdot X' = 0 \quad \text{and} \quad \dot{X}^2 + X'^2 = 0 $$

4. Quantization and the Critical Dimension

Solving the wave equation with Neumann boundary conditions (for open strings) yields a Fourier mode expansion of harmonic oscillators:

$$ X^\mu(\tau, \sigma) = x_0^\mu + 2\alpha' p_0^\mu \tau + i\sqrt{2\alpha'} \sum_{n \neq 0} \frac{\alpha_n^\mu}{n} e^{-in\tau} \cos(n\sigma) $$

When quantized, the zero-mode of the Hamiltonian (the $L_0$ Virasoro operator) becomes:

$$ L_0 = \frac{1}{2} \alpha_0^2 + \sum_{n=1}^\infty \alpha_{-n} \cdot \alpha_n + \epsilon_0 $$

Here, $\epsilon_0$ is the infinite zero-point vacuum energy of all oscillator modes across the $d-2$ transverse dimensions. Using Riemann zeta-function regularization ($\sum_{n=1}^\infty n = \zeta(-1) = -1/12$), we find:

$$ \epsilon_0 = \frac{d-2}{2} \sum_{n=1}^\infty n = -\frac{d-2}{24} $$

To preserve quantum conformal symmetry without anomalies, the intercept parameter $a$ must equal 1 ($a = -\epsilon_0 = 1$). Therefore:

$$ \frac{d-2}{24} = 1 \implies d = 26 $$

The mathematics strictly demands that Bosonic String Theory can only exist in a 26-dimensional spacetime!

The Tachyon Problem

The mass-squared operator for physical states is $m^2 = \frac{1}{\alpha'}(N - 1)$. For the ground state ($N=0$), we find $m^2 < 0$. This imaginary mass particle is the Tachyon. Its existence indicates that the bosonic vacuum is unstable, providing the historical and mathematical motivation to introduce fermions and Supersymmetry (leading to 10D Superstring Theory).

The Physics Network

Left-Click & Drag to Rotate | Scroll to Zoom | Click a Node to Fly