DOCSLIB.ORG

Basics of Qcd Perturbation Theory

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Basics of Qcd Perturbation Theory BASICS OF QCD PERTURBATION THEORY Davison E. Sop er Institute of Theoretical Science University of Oregon, Eugene, OR 97403 ABSTRACT This is an intro duction to the use of QCD p erturbation theory, em- phasizing generic features of the theory that enable one to separate short-time and long-time e ects. I also cover some imp ortant classes of applications: electron-p ositron annihilation to hadrons, deeply in- elastic scattering, and hard pro cesses in hadron-hadron collisions. Supp orted by DOE Contract DE-FG03-96ER40969. 1 Intro duction A prediction for exp eriment based on p erturbative QCD combines a particular calculation of Feynman diagrams with the use of general features of the theory. The particular calculation is easy at leading order, not so easy at next-to-leading order, and extremely dicult b eyond the next-to-leading order. This calculation of Feynman diagrams would b e a purely academic exercise if we did not use certain general features of the theory that allow the Feynman diagrams to b e related to exp eriment: the renormalization group and the running coupling; the existence of infrared safe observables; the factorization prop erty that allows us to isolate hadron structure in parton distribution functions. In these lectures, I discuss these structural features of the theory that allowa comparison of theory and exp eriment. Along the way,we will discover something ab out certain imp ortant pro cesses: + e e annihilation; deeply inelastic scattering; hard pro cesses in hadron-hadron collisions. By discussing the particular along with the general, I hop e to arm the reader with information that sp eakers at research conferences take to b e collective knowledge| knowledge that they assume the audience already knows. Now here is the disclaimer. We will not learn how to do signi cant calculations in QCD p erturbation theory. Three lectures is not enough for that. I hop e that the reader may b e inspired to pursue the sub jects discussed here 1 in more detail. A go o d source is the Handbook of Perturbative QCD by the CTEQ collab oration. More recently, Ellis, Stirling, and Webb er have written an 2 excellent book that covers most of the sub jects sketched in these lectures. For the reader wishing to gain a mastery of the theory, I can recommend the recent 3 4 5 b o oks on quantum eld theory by Brown, Sterman, Peskin and Schro eder, and 6 Weinb erg. Another go o d source, including b oth theory and phenomenology,is 7 the lectures in the 1995 TASI pro ceedings, QCD and Beyond. 2 Electron-Positron Annihilation and Jets In this section, I explore the structure of the nal state in QCD. I b egin with the + kinematics of e e ! 3 par tons, then examine the b ehavior of the cross section + for e e ! 3 par tons when two of the parton momenta b ecome collinear or one parton momentum b ecomes soft. In order to illustrate b etter what is going on, Iintro duce a theoretical to ol, null-plane co ordinates. Using this to ol, I sketch a space-time picture of the singularities that we nd in momentum space. The singularities of p erturbation theory corresp ond to long-time physics. We see that the structure of the nal state suggested by this picture conforms well with what is actually observed. I draw the distinction b etween short-time physics, for which p erturbation the- ory is useful, and long-time physics, for which the p erturbative expansion is out of control. Finally, I discuss how certain exp erimental measurements can prob e the short-time physics while avoiding sensitivity to the long-time physics. + 2.1 Kinematics of e e ! 3 Partons + Figure 1: Feynman diagram for e e ! q qg . p + Consider the pro cess e e ! q qg , as illustrated in Fig. 1. Let s b e the total energy in the c.m. frame and let q b e the virtual photon or Z b oson momentum, so q q = s. Let p b e the momenta of the outgoing partons q; q; g and let i 0 E = p b e the energies of the outgoing partons. It is useful to de ne energy i i fractions x by i 2p q E i i p = : 1 x = i s=2 s Then 0 <x : 2 i Energy conservation gives P X 2 p q i =2: 3 x = i s i Thus only two of the x are indep endent. i Let b e the angle b etween the momenta of partons i and j . We can relate ij these angles to the momentum fractions as follows: 2 2 2p p =p +p =qp =s2qp ; 4 1 2 1 2 3 3 2E E 1 cos =s1 x : 5 1 2 12 3 Dividing this equation by s=2 and rep eating the argument for the two other pairs of partons, we obtain three relations for the angles : ij x x 1 cos = 21 x ; 1 2 12 3 x x 1 cos = 21 x ; 2 3 23 1 x x 1 cos = 21 x : 6 3 1 31 2 We learn two things immediately. First, x < 1: 7 i Second, the three p ossible collinear con gurations of the partons are mapp ed into x space very simply: i ! 0 , x ! 1; 12 3 ! 0 , x ! 1; 23 1 ! 0 , x ! 1: 8 31 2 The relations 0 x 1, together with x =2x x, imply that the i 3 1 2 allowed region for x ;x is a triangle, as shown in Fig. 2. The edges x =1 of 1 2 i the allowed region corresp ond to two partons b eing collinear, as shown in Fig. 3. The corners x = 0 corresp ond to one parton momentum b eing soft p ! 0. i i 2.2 Structure of the Cross Section One can easily calculate the cross section corresp onding to Fig. 1 and the similar amplitude in which the gluon attaches to the antiquark line. The result is 2 2 d x + x 1 s 1 2 = C ; 9 F dx dx 2 1 x 1 x 0 1 2 1 2 Figure 2: Allowed region for x ;x . Then x is 2 x x . 1 2 3 1 2 Figure 3: Allowed region for x ;x . The lab els and small pictures show the 1 2 physical con guration of the three partons corresp onding to subregions in the allowed triangle. P 2 2 + where C =4=3 and =4 =s Q is the total cross section for e e ! F 0 f 0 hadr ons at order . The cross section has collinear singularities: s 1 x ! 0 ; 2&3 col l inear ; 1 1 x ! 0 ; 1&3 col l inear : 10 2 There is also a singularity when the gluon is soft: x ! 0. In terms of x and x , 3 1 2 this singularity o ccurs when 1 x 1 const: 11 1 x ! 0; 1 x ! 0; 1 2 1 x 2 Let us write the cross section in a way that displays the collinear singularity at ! 0 and the soft singularityat E ! 0: 31 3 d f E ; 1 s 3 31 = C : 12 F dE d cos 2 E 1 cos 0 3 31 3 31 Here, f E ; , a rather complicated function. The only thing that we need to 3 31 know ab out it is that it is nite for E ! 0 and for ! 0. 3 31 Now lo ok at the collinear singularity, ! 0.Ifweintegrate over the singular 31 region holding E xed, we nd that the integral is divergent: 3 Z 1 d d cos = log1: 13 31 dE d cos a 3 31 Similarly,ifweintegrate over the region of the soft singularity, holding xed, 31 we nd that the integral is divergent: Z a d = log1: 14 dE 3 dE d cos 0 3 31 Evidently, p erturbation theory is telling us that we should not take the p ertur- + bative cross section to o literally. The total cross section for e e ! hadr ons is certainly nite, so this partial cross section cannot b e in nite. What we are seeing is a breakdown of p erturbation theory in the soft and collinear regions, and we should understand why. + Figure 4: Cross section for e e ! q qg , illustrating the singularity when the gluon is soft or collinear with the quark. Where do the singularities come from? Lo ok at Fig. 4 in a physical gauge. 2 The scattering matrix element M contains a factor 1=p + p where 1 3 2 p + p =2p p =2E E 1 cos : 15 1 3 1 3 1 3 31 2 Evidently,1=p +p is singular when ! 0 and when E ! 0. The collinear 1 3 31 3 singularity is somewhat softened b ecause the numerator of the Feynman diagram contains a factor prop ortional to in the collinear limit. This is not exactly 31 obvious, but is easily seen by calculating. If you like symmetry arguments, you can derive this factor from quark helicity conservation and overall angular momentum conservation. Wethus nd that " 2 31 2 jMj / 16 2 E 3 31 for E ! 0 and ! 0. Note the universal nature of these factors. 3 31 Integration over the double singular region of the momentum space for the gluon has the form Z Z 2 E dE d cos d 3 31 3 2 E dE d d: 17 3 3 31 E 3 Combining the integration with the matrix element squared gives " 2 Z Z 2 dE d 3 31 31 2 d E dE d d d: 18 3 3 31 2 2 E E 3 3 31 31 Thus wehave a double logarithmic divergence in p erturbation theory for the soft and collinear region.
Load more

Recommended publications
  • How to Hack a Turned-Off Computer Or Running Unsigned
    HOW TO HACK A TURNED-OFF COMPUTER, OR RUNNING UNSIGNED CODE IN INTEL ME Contents Contents ................................................................................................................................ 2 1. Introduction ...................................................................................................................... 3 1.1. Intel Management Engine 11 overview ............................................................................. 4 1.2. Published vulnerabilities in Intel ME .................................................................................. 5 1.2.1. Ring-3 rootkits.......................................................................................................... 5 1.2.2. Zero-Touch Provisioning ........................................................................................... 5 1.2.3. Silent Bob is Silent .................................................................................................... 5 2. Potential attack vectors ...................................................................................................... 6 2.1. HECI ............................................................................................................................... 6 2.2. Network (vPro only)......................................................................................................... 6 2.3. Hardware attack on SPI interface ..................................................................................... 6 2.4. Internal file system .........................................................................................................
    [Show full text]
  • Issue #63, July 2000 Starting Our SIXTH Year in Publishing!
    Issue #63, July 2000 Starting our SIXTH year in publishing! 64a Page 1 Wed, Jul 2000 Cover by: Bill Perry [email protected] Published by My Mac Productions 110 Burr St., Battle Creek, MI 49015-2525 Production Staff Tim Robertson • [email protected] Publisher / Creator / Owner Editor-in-Chief Adam Karneboge • [email protected] Webmaster / Contributing Editor Roger Born • [email protected] Website Edior Barbara Bell • [email protected] Director, Public Relations •Jobs & Woz • Inspiration Artwork Created by: •Mike Gorman• [email protected] •Bill Perry• [email protected] •Tim Robertson• [email protected] •Adam Karneboge• [email protected] This Publication was created with: DOCMaker v4.8.4 http://www.hsv.tis.net/~greenmtn & Adobe Acrobat 4.0 http://www.adobe.com 64a Page 2 Wed, Jul 2000 Other Tools: Adobe Photoshop 5.5, 5.0.1 ColorIt! 4.0.1 BBEdit Lite ClarisWorks 5.0 Microsoft Word 98 GraphicConverter Snapz Pro 2.0 SimpleText Netscape Communicator 4.6.1 Internet Explorer 4.5 Eudora Pro 4.0.2 FileMaker Pro 4.0v3 QuickKeys 4.0 and the TitleTrack CD Player (To keep us sane!) Website hosted by Innovative Technologies Group Inc. http://www.inno-tech.com My Mac Magazine ® 1999-2000 My Mac Productions. All Rights Reserved. 64a Page 3 Wed, Jul 2000 http://www.inno-tech.com http://www.smalldog.com http://www.megamac.com 64a Page 4 Wed, Jul 2000 Advertising in My Mac = Good Business Sense! With over 500,000 website visits a month and thousands of email subscribers, You just can't go wrong! Send email to [email protected] for information.
    [Show full text]
  • Intel Management Engine Deep Dive
    Intel Management Engine Deep Dive Peter Bosch About me Peter Bosch ● CS / Astronomy student at Leiden University ● Email : [email protected] ● Twitter: @peterbjornx ● GitHub: peterbjornx ● https://pbx.sh/ About me Previous work: ● CVE-2019-11098: Intel Boot Guard bypass through TOCTOU attack on the SPI bus (Co-discovered by @qrs) Outline 1. Introduction to the Management Engine Operating System 2. The Management Engine as part of the boot process 3. Possibilities for opening up development and security research on the ME Additional materials will be uploaded to https://pbx.sh/ in the days following the talk. About the ME About ME ● Full-featured embedded system within the PCH ○ 80486-derived core ○ 1.5MB SRAM ○ 128K mask ROM ○ Hardware cryptographic engine ○ Multiple sets of fuses. ○ Bus bridges to PCH global fabric ○ Access to host DRAM ○ Access to Ethernet, WLAN ● Responsible for ○ System bringup ○ Manageability ■ KVM ○ Security / DRM ■ Boot Guard ■ fTPM ■ Secure enclave About ME ● Only runs Intel signed firmware ● Sophisticated , custom OS ○ Stored mostly in SPI flash ○ Microkernel ○ Higher level code largely from MINIX ○ Custom filesystems ○ Custom binary format ● Configurable ○ Factory programmed fuses ○ Field programmable fuses ○ SPI Flash ● Extensible ○ Native modules ○ JVM (DAL) Scope of this talk Intel ME version 11 , specifically looking at version 11.0.0.1205 Platforms: ● Sunrise Point (Core 6th, 7th generation SoC, Intel 100, 200 series chipset) ● Lewisburg ( Intel C62x chipsets ) Disclaimer ● I am in no way affiliated with Intel Corporation. ● All information presented here was obtained from public documentation or by reverse engineering firmware extracted from hardware found “in the wild”. ● Because this presentation covers a very broad and scarcely documented subject I can not guarantee accuracy of the contents.
    [Show full text]
  • The Strangeness Magnetic Moment of the Proton in the Chiral Quark Model
    The Strangeness Magnetic Moment of the Proton in the Chiral Quark Model L. Hannelius, D.O. Riska Department of Physics, University of Helsinki, 00014 Finland and L. Ya. Glozman Institute for Theoretical Physics, University of Graz, A-8019 Graz, Austria Abstract The strangeness magnetic moment of the proton is shown to be small in the chiral quark model. The dominant loop contribution is due to kaons. The K∗ loop contributions are proportional to the difference between the strange and light constituent quark masses or −2 mK∗ and therefore small. The loop fluctuations that involve radiative transitions between K∗ mesons and kaons are small, when the cut-off scale in the loops is taken to be close to the chiral symmetry restoration scale. The net loop amplitude contribution to the strangeness magnetic moment of the proton is about −0.05, which falls within the uncertainty range of arXiv:hep-ph/9908393v2 24 Aug 1999 the experimental value. 0 1. Introduction The recent finding by the SAMPLE collaboration that the strangeness magnetic moment s 2 2 of the proton is small, and possibly even positive [1] (GM (Q = 0.1 GeV )=0.23 ± 0.37) was unexpected in view of the fact that the bulk of the many theoretical predictions for this quantity are negative, and outside of the experimental uncertainty range (summaries are given e.g. in refs. [2, 3, 4]). A recent lattice calculation gives −0.36 ± 0.20 for this quantity [5], thus reaffirming the typical theoretical expectation, while remaining outside of the uncertainty range of the empirical value.
    [Show full text]
  • An Introduction to Morphos
    An Introduction to MorphOS Updated to include features to version 1.4.5 May 14, 2005 MorphOS 1.4 This presentation gives an overview of MorphOS and the features that are present in the MorphOS 1.4 shipping product. For a fully comprehensive list please see the "Full Features list" which can be found at: www.PegasosPPC.com Why MorphOS? Modern Operating Systems are powerful, flexible and stable tools. For the most part, if you know how to look after them, they do their job reasonably well. But, they are just tools to do a job. They've lost their spark, they're boring. A long time ago computers were fun, it is this background that MorphOS came from and this is what MorphOS is for, making computers fun again. What is MorphOS? MorphOS is a fully featured desktop Operating System for PowerPC CPUs. It is small, highly responsive and has very low hardware requirements. The overall structure of MorphOS is based on a new modern kernel called Quark and a structure divided into a series of "boxes". This system allows different OS APIs to be used along side one another but isolates them so one cannot compromise the other. To make sure there is plenty of software to begin with the majority of development to date has been based on the A- BOX. In the future the more advanced Q-Box shall be added. Compatibility The A-Box is an entire PowerPC native OS layer which includes source and binary compatibility with software for the Commodore A500 / A1200 etc.
    [Show full text]
  • Basics of Qcd Perturbation Theory
    BASICS OF QCD PERTURBATION THEORY Davison E. Soper* Institute of Theoretical Science University of Oregon, Eugene, OR 97403 ABSTRACT (•• i This is an introduction to the use of QCD perturbation theory, em- I phasizing generic features of the theory that enable one to separate short-time and long-time effects. I also cover some important classes of applications: electron-positron annihilation to hadrons, deeply in- elastic scattering, and hard processes in hadron-hadron collisions. •Supported by DOE Contract DE-FG03-96ER40969. © 1996 by Davison E. Soper. -15- 1 Introduction 2 Electron-Positron Annihilation and Jets A prediction for experiment based on perturbative QCD combines a particular In this section, I explore the structure of the final state in QCD. I begin with the calculation of Feynman diagrams with the use of general features of the theory. kinematics of e+e~ —> 3 partons, then examine the behavior of the cross section The particular calculation is easy at leading order, not so easy at next-to-leading for e+e~ —i- 3 partons when two of the parton momenta become collinear or one order, and extremely difficult beyond the next-to-leading order. This calculation parton momentum becomes soft. In order to illustrate better what is going on, of Feynman diagrams would be a purely academic exercise if we did not use certain I introduce a theoretical tool, null-plane coordinates. Using this tool, I sketch general features of the theory that allow the Feynman diagrams to be related to a space-time picture of the singularities that we find in momentum space.
    [Show full text]
  • Introduction to Storage and Software Systems for Data Analysis
    INTRODUCTION TO STORAGE AND SOFTWARE SYSTEMS FOR DATA ANALYSIS Bob Jacobsen University of California, Berkeley, USA Abstract The Storage and Software Systems for Data Analysis track discusses how HEP physics data is taken, processed and analyzed, with emphasis on the problems that data size and CPU needs pose for people trying to do experimental physics. The role of software engineering is discussed in the context of building large, robust systems that must at the same time be accessible to physicists. We include some examples of existing systems for physics analysis, and raise some issues to consider when evaluating them. This lecture is the introduction to those topics. 1. INTRODUCTION Many modern high-energy physics (HEP) experiments are done by collaborations of hundreds of people. Together, these groups construct and operate complex detectors, recording billions of events and terabytes of data, all toward the goal of “doing physics”. In this note, we provide an introduction to how we currently do this, and raise a number of issues to be considered when thinking about the new systems that are now being built. 2. SCALE OF THE EXPERIMENTS BaBar, CDF and D0 are examples of the large experiments now taking or about to take data in a collider environment. The collaborations that have built these experiments contain 300 to 600 members with varying levels of activity. Almost everybody is considered “familiar” with using computing to do their work, but only a small fraction of the collaboration can be considered as computing professionals. Some of these can even be considered world-class experts in large scale computing.
    [Show full text]
  • Quarkxpress 8.0 Readme Ii
    QuarkXPress 8.0 ReadMe ii Contents QuarkXPress 8.0 ReadMe....................................................................................................3 System requirements.............................................................................................................4 Mac OS.....................................................................................................................................................4 Windows...................................................................................................................................................4 Installing: Mac OS................................................................................................................5 Performing a silent installation.................................................................................................................5 Preparing for silent installation....................................................................................................5 Installing.......................................................................................................................................5 Performing a drag installation..................................................................................................................5 Adding files after installation...................................................................................................................6 Installing: Windows..............................................................................................................7
    [Show full text]
  • Lepton Probes in Nuclear Physics
    L C f' - l ■) aboratoire ATIONAI FR9601644 ATURNE 91191 Gif-sur-Yvette Cedex France LEPTON PROBES IN NUCLEAR PHYSICS J. ARVIEUX Laboratoire National Saturne IN2P3-CNRS A DSM-CEA, CESaclay F-9I191 Gif-sur-Yvette Cedex, France EPS Conference : Large Facilities in Physics, Lausanne (Switzerland) Sept. 12-14, 1994 ££)\-LNS/Ph/94-18 Centre National de la Recherche Scientifique CBD Commissariat a I’Energie Atomique VOL LEPTON PROBES IN NUCLEAR PHYSICS J. ARVTEUX Laboratoire National Saturne IN2P3-CNRS &. DSM-CEA, CESaclay F-91191 Gif-sur-Yvette Cedex, France ABSTRACT 1. Introduction This review concerns the facilities which use the lepton probe to learn about nuclear physics. Since this Conference is attended by a large audience coming from diverse horizons, a few definitions may help to explain what I am going to talk about. 1.1. Leptons versus hadrons The particle physics world is divided in leptons and hadrons. Leptons are truly fundamental particles which are point-like (their dimension cannot be measured) and which interact with matter through two well-known forces : the electromagnetic interaction and the weak interaction which have been regrouped in the 70's in the single electroweak interaction following the theoretical insight of S. Weinberg (Nobel prize in 1979) and the experimental discoveries of the Z° and W±- bosons at CERN by C. Rubbia and Collaborators (Nobel prize in 1984). The leptons comprise 3 families : electrons (e), muons (jt) and tau (r) and their corresponding neutrinos : ve, and vr . Nuclear physics can make use of electrons and muons but since muons are produced at large energy accelerators, they more or less belong to the particle world although they can also be used to study solid state physics.
    [Show full text]
  • Phenomenological Review on Quark–Gluon Plasma: Concepts Vs
    Review Phenomenological Review on Quark–Gluon Plasma: Concepts vs. Observations Roman Pasechnik 1,* and Michal Šumbera 2 1 Department of Astronomy and Theoretical Physics, Lund University, SE-223 62 Lund, Sweden 2 Nuclear Physics Institute ASCR 250 68 Rez/Prague,ˇ Czech Republic; [email protected] * Correspondence: [email protected] Abstract: In this review, we present an up-to-date phenomenological summary of research developments in the physics of the Quark–Gluon Plasma (QGP). A short historical perspective and theoretical motivation for this rapidly developing field of contemporary particle physics is provided. In addition, we introduce and discuss the role of the quantum chromodynamics (QCD) ground state, non-perturbative and lattice QCD results on the QGP properties, as well as the transport models used to make a connection between theory and experiment. The experimental part presents the selected results on bulk observables, hard and penetrating probes obtained in the ultra-relativistic heavy-ion experiments carried out at the Brookhaven National Laboratory Relativistic Heavy Ion Collider (BNL RHIC) and CERN Super Proton Synchrotron (SPS) and Large Hadron Collider (LHC) accelerators. We also give a brief overview of new developments related to the ongoing searches of the QCD critical point and to the collectivity in small (p + p and p + A) systems. Keywords: extreme states of matter; heavy ion collisions; QCD critical point; quark–gluon plasma; saturation phenomena; QCD vacuum PACS: 25.75.-q, 12.38.Mh, 25.75.Nq, 21.65.Qr 1. Introduction Quark–gluon plasma (QGP) is a new state of nuclear matter existing at extremely high temperatures and densities when composite states called hadrons (protons, neutrons, pions, etc.) lose their identity and dissolve into a soup of their constituents—quarks and gluons.
    [Show full text]
  • Inside Track
    Inside Track JOHN C. DVORAK neaking In the Back Door Dept.: to use those pesky legacy apps. A European If you think the battle of plat- If this machine achieves critical mass, consortium forms involves only Linux, it could reignite the dull computer Mac, and Windows, you’re in scene. There is a weird—even freaky— wants to float Sfor a rude awakening. enthusiasm about this project. It’s a a slew of Since 1995, a group of superstars work- community-based movement, which blimps to deliv- ing primarily in Europe has literally means that everything’s free. Tools are er broadband. morphed the architecture of the we-all- free, the OS is free, applications that thought-it-was-dead AmigaOS, probably cost money for other OSs are free for Who says the slickest, tightest code ever put on a MorphOS. The geeks are in high gear. they don’t desktop computer. With seven years of This is much more exciting than the smoke potent clean-room engineering, the new Morph- pedantic Linux world, which can’t seem OS runs on the PowerPC chip, and literal- to consolidate, let alone compete with marijuana in ly thousands of developers worldwide are Microsoft on the desktop. Europe? working with it. The new OS is best It’s not impossible for a new platform showcased in the PowerPC-based Genesi to emerge. Remember how Nintendo works. SkyLinc, from York, England, Pegasos computer, which runs both Mor- and Sega dominated the game scene for wants to position tethered blimps 5,000 phOS and various PPC versions so many years after the collapse of feet in the air over rural areas (with mile- of Linux.
    [Show full text]
  • MINIX 3 Di Alessandro Bottoni
    Mensile edito dall'Associazione di promozione sociale senza scopo di lucro Partito Pirata G Iscrizione Tribunale di Rovereto Tn n° 275 direttore responsabile Mario Cossali p.IVA/CF01993330222 I U G N O 2 0 0 8 anno 1 numero 7 prezzo di vendita: OpenContent (alla soddisfazione del lettore) L’IMMAGINE DEL MINORE E LA SUA TUTELA di Valeria Falcone 7 MINIX 3 di Alessandro Bottoni L'ossimoro dell' anonimato libero La Porta Galleggiante di Miyajima di Alessandro Bottoni Per parlare di anonimato è necessario prima chia- ecc.), lo fa per due motivi principali: 1) denuncia- rire cosa s'intende per un termine ben più re una vicenda che dovrebbe essere conosciuta pervasivo: libertà. Possiamo definire libero il dal pubblico e dalle Forze dell'ordine senza te- pensiero umano, ciò che pensiamo è uno ma di subire ritorsioni a livello personale; 2)non scambio di neuroni confinati all'interno della no- subire alcuna limitazione per la propria espres- stra scatola cranica, dipende da noi tradurli in sione. In entrambi i casi si tratta comunque di parole o lasciarli confinati nel nostro cervello. Al mancanza di libertà, nel primo caso c'è la paura di là del pensiero la libertà scema influenzata da delle conseguenze derivanti da una condizione migliaia di variabili, ambientali, culturali o situazio- oggettiva di inferiorità vera o supposta, nel se- ni contingenti. In qualsiasi società la libertà è poi condo caso per condizionamenti, i più diversi, o limitata dalle regole che permettono di definire per evitare coinvolgimenti personali. In una so- più o meno marcatamente la libertà dei singoli in cietà idilliaca ogni cittadino dovrebbe essere rapporto alla collettività.
    [Show full text]